Partition of Unity Based Shell Finite Elements

Partition of Unity Based Shell Finite Elements

1

Hyungmin Jun

Ocean Systems Engineering, KAISTNov. 4th, 2014

Ph. D. Thesis Presentation

2

Goal

: To develop the efficient and robust shell finite elements based on partition of unity concept.

• Linear and nonlinear static analyses• Shell elements with the effects shear deformation (Reissner-Mindlin theory)

Committee

Prof. PhillSeung Lee, Chair (Ocean Systems Engineering, KAIST) Prof. HyukChun Noh (Civil and Environmental Engineering, Sejong University) Prof. Hyun Chung (Ocean Systems Engineering, KAIST) Prof. YeunWoo Cho (Ocean Systems Engineering, KAIST) Prof. Seunghwa Ryu (Mechanical Engineering, KAIST)

Limitations :

Ph. D. Thesis Presentation

3

Contents : Partition of unity based shell finite elements

1. Part I (Introduction to Partition of Unity (PU) Based FEM) Brief History of the Finite Element Method Application of the Finite Element Method Finite Element Analysis Procedures Ultimate Goal of the Finite Element Method Partition of Unity Based Shell Finite Element

2. Part II (PU Based Shell Element with Improved Membrane Behaviors) Introduction to Nonlinear Finite Element Analysis Introduction and Scope of Research MITC3+ in the Nonlinear Analysis

Key Concepts / Nonlinear Formulation Benchmark Problems

The Method with Improved Membrane Behaviors Comparison with Other Methods Key Concepts / Nonlinear Formulation Benchmark Problems

3. Future Works

4. Conclusions

Enriched MITC3

MITC3+

Enriched MITC3+

4

Part I:

Introduction to Partition of Unity (PU) Based FEM

Brief History of the Finite Element Method Application of the Finite Element Method Finite Element Analysis Procedures Ultimate Goal of the Finite Element Method Partition of Unity Based Shell Finite Elements

5

Brief history – Events in the Finite Element Methods

Events in the Finite Element Methods

1960 1970 1980 1990 2000 2010

1956: First paper by Turner, Clough, Martin and Topp

1960: FEM coined by Clough

1970s: Commercial software(ABAQUS, ANSYS, ADINA)

1982: ANS by MacNeal

1976: Incompatible modeby Taylor

2000: GFEM (PU FEM)by Strouboulis

UC Berkely 1950

1958: UC program using IBM 704 by Wilson

1967: First book by Zienkiewicz

1971: Reduced integrationby Zienkiewicz

1971: First FEAPby Taylor

1990: EAS by Simo

1984: MITCby Bathe

2005: IGAby Hughes

1999: XFEM by Belytschko

1962: Isoparametric element by Taig

1966: Numerical integration by Irons

1970: Curved shell by Ahmad

~1960• Direct Stiffness Method• Force Method

6

FEA software list

Commercial (48)ABAQUSANSYSADINANASTRANALGORLS-DYNAPAM-CRASHMARCLUSASCOMSOLVISUALFEAFEMAP……..

Open source (19)FreeFem++OOFEMjFEMGetFEM++Code Aster…….. Commercial FEA software prices start from around $1,500 to $60,000.

Proprietary / Commercial Finite Element Analysis (FEA) Software

List of finite element software packages – Commercial FEA software

ADINAABAQUS

VISUALFEA

ANSYS

ALGOR NASTRAN

Figure source : google image

7

Automobile Aerospace Electronics Civil engineering

Healthcare Sports Defense Ship building

Applications of FEM #1 – Industry fields


8

Olivier et al., Science 2008(Developmental patterning)

Dario et al., Nature Communications 2011(Frequency stabilization)

Diederik et al. Nature Photonics 2013(Disordered photonics)

McHenry et al. PNAS 2007(Predatory behavior in Smilodon)

Giorgio et al., Nature Communications 2012(Microscopic magnetic simulation)

Sampathkumar et al. Cell Current Biology 2014(Plant development)

Anderson et al. Cell Theory 2006(Tumor morphology)

Masson et al. PNAS 2006(Ionic contrast terahertz)

Applications of FEM #2 – Research fields (High profile journal)

Figure source : journals

9

A Study on the Finite Element Method

Of course, more research is neededdespite the remarkable developments and improvements.

ANSWER :

Many researchers and their contributions over 50 years

More than 10,000,000 papers to improve performance of the FEMs

Most widely used numerical methods in engineering

Stabilized commercial software products (e.g. ABQUS, ANSYS, ADINA,…)

Is there nothing left to improve FEMs?CLUE

10

FE analysis procedures – Step 1: Observation of the system

Step 1: Observation of the system

This eccentric load is perpendicular to the column, but it doesn’t pass through the column’s centroid.

Column under an eccentric compressive load


11

FE analysis procedures – Step 2: Idealization of the system

Step 2: Idealization of the system

Major assumption : Linear elastic material

Imposing the boundary condition

Appling an eccentric load

Idealization

See the dissertation, p. 121

12

FE analysis procedures – Step 3: Discretization of the domain

Step 3: Discretization of the domain (using ADINA founded in 1974, ADINA 9.0.5 latest released version (2014.10))(setting the material properties, imposing the boundary condition, and applying loads)

FE discretization

13

FE analysis procedures – Step 4: Solving the problem and interpreting the result

Step 4: Solving the problem and interpreting the result

Result : deformed shape of the column

More accurate result


14

The cause of this problem – In-plane shear locking

In-plane shear locking : The element has an excess of shear strain which contributes to the poor ability of the element to reproduce bending modes.

2D pure bending problem 2 2 2 2

2 2( , ) 1 12 2Ma r Mb sv r s

EI EIa b

( , ) Mu r s rsEI

0xyu vy x

Correct beam deformation

( , )u r s urs

0v

xy ur

Parasitic in-plane shear strain

Finite element deformation


15

Remedies – Books and manual books

Amazon search450 results in science & math, keyword : “finite element method”

Commercial software : ADINAManual book : 12 books

16

The ultimate goal of the finite element method #1 – User conveniences & Accuracy

Accurate solution & User conveniences

One click Reliable results

Controllability

What does “user conveniences” mean?

17

The ultimate goal of the finite element method #2 – User conveniences & Accuracy

Step 2: Idealization of the system

Step 3: Discretization of the domain

Step 4: Solving the problem

Step 1: Observation of the systemG

ener

al p

roce

dure

Add

ition

al p

roce

dure

Estimating the FE errors

Locally enrichment without mesh refinements

Accurate solution


18

The ultimate goal of the finite element method #3 – Error estimates

“A Finite element solution contains enough information to estimate its own error”

Cook RD et al., Concepts and applications of finite element analysis, 4th ed., Wiley.

Two types of error estimates serve very different purpose

A priori error estimates To check the order of convergence of a given FE method

A posteriori error estimates To indicate where the error is particularly high

I. Residuals – based methodBabuska I and Rheinboldt WC, Int. J. Numer. Meth. Eng.;12:1597-615, 1978 (1371 cited)

II. Recovery – based methodZienkiewiz OC and Zhu JZ, Int. J. Numer. Meth. Eng.;24:337-57,1987 – ZZ method (2147 cited)

“Nowadays, a booming activity in error estimation is fostered by better understanding of mathematical foundations”

Aninsworth M, Oden JT, A posteriori error estimation in finite element analysis, Wiley

Without mesh refinements

Without using transition elements

p-refinement >> h-refinement

Shell elements alleviating locking

Combining with adaptive mesh refinement

Partition of unity based shell finite element(Enriched MITC3 shell element)

19


Hyang et al., DDMSE;78:53-74,2011

Error estimate / Adaptive mesh refinement

Disadvantages

Mesh refinement algorithms Only h-refinement 2D / 3D solid problem Transition elements

If shell problem,how can treat the locking phenomenon?


5-nodeelement

4-nodeelement

4-nodeelement

20


Hyang et al., DDMSE;78:53-74,2011

Error estimate / Adaptive mesh refinement

Disadvantages

Mesh refinement algorithms Only h-refinement 2D / 3D solid problem Transition elements

If shell problem,how can treat the locking phenomenon?


5-nodeelement

4-nodeelement

4-nodeelement

21

3

1( ) i i

iu h u

xFEM

3

1( ) ( )i i

iu h u

x xPU based FEM : local approximation function( )iu x

ih : shape function

2 3( ) ( ) ( )i i i i i iu u x x a y y a x

3

1

3

2 3

3

21

1

13

3

( )

( ) ( )

( ) ( )

( )

ˆ

i ii

i i i i i i

i ii i i ii i

i

ii

u h u

h u

h x x a h y y a

x x a y y a

h u

uu

x x

: nodal displacement variableiu

T( ) ( )iu x p a x

Local approximation ,

T ( ) [1, , , , ]x y xyp x

1 2 3 4( ) [ , , , , ]i i i ia a a aa x

1 2 3( , )i i i i i i i i iu x y a x a y a u

1 2 3i i i i i ia u x a y a

1 2 3( )i i i iu a xa ya x

( )iu x

PU based Finite Element Method – Displacement interpolation


Use of the high order interpolation Local use

22

, ,i , ,i1 1 ˆ ˆ2 2ij i j j i j je g u g u g u g u

, ,i12ij i j je g u g u

The linear terms of the enriched covariant strain components

MITC3

1

2

rt

st

e a cs

e a cr

ˆ( )i i

i ir r

x x xg g

PU based MITC3 shell element – Locking treatment

MITC6

1 1 1

2 2 2

rt

st

e a b r c s

e a b r c s

Convergence study

# of elements

Dec

reas

e of

the

erro

rs

PU based MITC3 shell element(also called enriched MITC3)

23

Jeon HM, Lee PS, Bathe KJ. The MITC3 shell finite element enriched by interpolation covers. Comput Struct 2014;134:128-42.

Fine mesh

Coarse mesh

PU based MITC3 shell element – “Highly sensitive” shell problem

24

Journal paper – The MITC3 shell finite element enriched by interpolation covers

25

The ultimate goal of the FEMs – PU based shell finite elements

Locally enriched by the PU based shell elements

Conventional FEM processes

Error estimation

Reliable results

Con

trolla

ble

Fini

te E

lem

ent M

etho

d

BiologistEngineerChemistry Analysis of nan

Biological phenEngineering stru

Black box

26

Part II:

PU based Shell Element with Improved Membrane Response)

Introduction to Nonlinear Finite Element Analysis Introduction and Scope of Research MITC3+ Shell Element in the Nonlinear Analysis

Key Concepts / Nonlinear Formulation Benchmark Problems

The Method with Improved Membrane Behaviors Comparison with Other Methods Key Concepts / Nonlinear Formulation Benchmark Problems

27

• Geometric nonlinear analysis• Material nonlinear analysis• Changing in boundary condition

Classification

Linear analysis Nonlinear analysisCantilever beam subjected to end shear force

Introduction to nonlinear FEA #1 – Linear analysis vs Nonlinear analysis

4

1, 10,1.2 10 , 0.34.5

b h lE vP

28

Introduction to nonlinear FEA #2 – Linear analysis vs Nonlinear analysis

LINEAR ANALYSIS NONLINEAR ANALYSIS

1. Strain Infinitesimal strain 12

jiij

j i

uue

x x

Green-Lagrange strain 12

k k

i j

jiij

j i

uue

x xxu u

x

2. Rotation (curved beam/shell)

Infinitesimal rotation t t i t t i t i

n t n V Q V

t t i t t it 3 t

Q I Θ

3 2

3 1

2 1

00

0

t t i t t it t

t t i t t i t t it t t

t t i t t it t

Θ

▶ 2 1i i

i i V V

Finite rotation t t i t t i t i

n t n V Q V

t t i t t i t t i 2t 3 t t

1 ( )2!

Q I Θ Θ

3 2

3 1

2 1

00

0

t t i t t it t

t t i t t i t t it t t

t t i t t it t

Θ

▶ 22 2

11 ( )2

t t ii i n

i t ii i V V V

3. Framework

-

◇ Total Lagrangian formulation ◇ Update Lagrangian formulation ◇ Corotational formulation

4. Iterative scheme

-

○ Full Newton-Raphson method ○ Modified Newton-Raphson method ○ BFGS matrix update method

29

Three types of approaches : to improve the nonlinear finite element solution

Introduction to nonlinear FEA #3 – Approaches to improve nonlinear finite element scheme

1. Using the high performance computing

2. Obtaining the Equilibrium path

3. Development and improvement of the finite elements (present research)

30

Read geometric / material information

Assemble loads

Calculation of stresses

Solving the linear equation( 1) ( ) ( 1)

0 0t i i t t t i K U R F

Update position vector( ) ( 1) ( )t t i t t i i U U U

Assemble internal force vector

0

T 00 0t t ij

ijVd V F B S

Assemble tangential stiffness matrix

0

T 00 0t ijkl

L ij klVC d V K B B

0

00t t ij

NL ij oVS d V K N

End program

Start program

New

ton

Rap

hson

pro

cedu

re

No

No Inc. loads

Convergence

Incr

emen

tal l

oad

step

pro

cedu

reGeometric nonlinear FE procedure

Three types of approaches #1 – High performance computing

Two quad core Intel Xeon X5560Core : 8 running at 3.2 GHzRAM : 48 GB of DDR31333MHzParallel library : OpenMP

NVIDIA GeForce GTX480Core : 480 CUDA, 1401 MHzRAM : 1.5 GB of video memoryParallel library : CUDA

Dick et al., Simul Model Pract Th 2011;19:801-16.Karatarakis et al., Comput Methods Appl Mech Engrg 2014;269:334-55.

High performance computing

31


Assemble loads






0

T 00 0t t ij

ijVd V F B S


0

T 00 0t ijkl

L ij klVC d V K B B

0

00t t ij

NL ij oVS d V K N

End program

Start program

New

ton

Rap

hson

pro

cedu

re

No

No Inc. loads

Convergence

Incr

emen

tal l

oad

step

pro

cedu


Load control / Displacement control (Batoz and Dhatt, 1979) Limit point analysis – Arc length method

(Riks(1972), Crisfiled(1980)) Others – Automatic load step(Ramm, 1982), Line search(Box 1969)

Obtaining the equilibrium path

Three types of approaches #2 – Obtaining the equilibrium path

32


Assemble loads






0

T 00 0t t ij

ijVd V F B S


0

T 00 0t ijkl

L ij klVC d V K B B

0

00t t ij

NL ij oVS d V K N

End program

Start program

New

ton

Rap

hson

pro

cedu

re

No

No Inc. loads

Convergence

Incr

emen

tal l

oad

step

pro

cedu


S3, S3R, S4, S4R, SAX1, SAX2, SAX2T, SAXA1n, SAXA2n, STRI3, S4R5, STRI65, S8R, S8RT, S8R5, S9R5

SHELL28, SHELL41, SHELL43, SHELL51, SHELL57, SHELL61SHELL63, SHELL91, SHELL93, SHELL99, SHELL143, SHELL150SHELL157, SHELL163, SHELL181, SHELL185

Axisymmetric shell, DISP10 Collapsed MITC3, MITC4, MITC4i, MITC6, MITC8, MITC9, MITC16

QUAD4, QUAD8, QUADR, TRIA3, TRIA6, TRIAR, TRIA, CONEAX, RTRPLT

Different types of shell elements in commercial software

Three types of approaches #3 – Development and improvement of the finite elements

33

Validation of the nonlinear FE code – CODE vs ADINA

Commercial software (ADINA) CODE (C / C++ / Fortran)

MITC4 shell element Full Newton-Raphson method 100 load step

Deformed shapes• CODE (C++)• TECPLOT / PARAVIEW

34

Shell element = membrane element + plate element

Locking problemsin 3-node shell elements

Membrane locking

Transverse shear locking


• In-plane shear locking• Trapezoidal locking• Volumetric locking

In-plane shear locking Trapezoidal locking

Introduction to locking phenomenon – 3-node triangular shell element

35

Newly developed shell elements – MITC3+ and enriched MITC3+ shell elements

Two types of newly developed triangular shell elements

Membrane locking


In-plane shear / Trapezoidal / Volumetric locking

MITC3+PU based MITC3+(Enriched MITC3+)

MITC3+ Enriched MITC3+

1. BASIS

Original MITC3 shell element

MITC3+ shell element

2. METHODS

Cubic bubble function


Partition of unity approximation

4. FEATURE

Improved bending behaviors

Improved bending / membrane behaviors

36

MITC3+ shell element – Key concepts

Lee Y et al., The MITC3+ shell element and its performance, Comput Struct 2014;138:12-23 Excellent convergence behavior Only linear analysis conditions have been considered

3 3

2 11 1

( , , ) ( , ) ( , )( )2

i ii i i i i i

i i

r s h r s a h r s

u u V V-∑ ∑

3 4

2 11 1

( , , ) ( , ) ( , )( )2

i ii i i i i i

i i

r s h r s a f r s

u u V V-∑ ∑

Original 3-node shell element

3-node shell element with the bubble function

Suggested new assumed strain fields MITC3+ shell element

37

MITC3+ shell element – Nonlinear formulation

Incremental Green-Lagrange strain tensor components :

Displacement interpolation with finite rotations :

3 42 2

2 11 1

1( , ) ( )2 2

t i t i t ii i i i i i i i n

i i

th r s a f

u u V V V

0 , , , ,1 1( ) ( )2 2

t t t t t t t tij i j i j i j i j i jε g g g g u g g u u u

Kinematics

nonlinear part in the GL strain

Iterative solution procedures

( 1) ( ) ( 1)0 0t i i t t t i K U F

)()1()( iittitt UUU

With the full Newtorn-Raphson iteration scheme, the equation for the i-th iteration in a finite element model are

for the displacement,

3 4

2 11 1 2

i ii i i i i i

i i

th a f

u u V Vcf. linear analysis

38

Nonlinear performances of the MITC3+ element #1 – Slit annular plate problem

Slit annular plate problem

6

6, 10, 0.03, 3.2

21 10 , 0i eR R h P

E v

Element Description

MITC4 4-node quadrilateral shell element(Bathe, 1986)

MITC3 3-node triangular shell element(Lee, 2004)

MITC3+ 3-node triangular shell elementwith a cubic bubble function (present)

Reference 9-node quadrilateral shell element

39

Hemisphere shell problem

Nonlinear performances of the MITC3+ element #2 – Hemisphere shell

7

10, 0.04, 4006.825 10 , 0.3

R h PE v

Element Description



MITC3+ 3-node triangular shell elementwith a cubic bubble function (present)


40

Plate under end shear force Plate subjected to end moment Clamped semi-cylindrical shell

Nonlinear performances of the MITC3+ element #3 – Other problems

Other benchmark problems solved

41

Newly developed shell elements – MITC3+ and enriched MITC3+ shell elements

Two types of newly developed triangular shell elements

Membrane locking


In-plane shear / Trapezoidal / Volumetric locking

MITC3+PU based MITC3+(Enriched MITC3+)

MITC3+ Enriched MITC3+

1. BASIS

Original MITC3 shell element


2. METHODS




4. FEATURE

Improved bending behaviors

Improved bending / membrane behaviors

42

In plane shear locking – Bilinear quadrilateral finite element

2 2 2 2

2 2( , ) 1 12 2Ma r Mb sv r s

EI EIa b

( , ) Mu r s rsEI

( , )u r s urs

0v

0xyu vy x

xy ur

Parasitic in-plane shear strain

In-plane shear locking : The element has an excess of shear strain which contribute to the poor ability of the element to reproduce bending modes.

Pure bending 2D problem

Correct beam deformation

Finite element deformation

43

Comparison with other methods #1 – Introductions

The methods to improve membrane behaviors

Reduced Integration method Incompatible mode Additional bubble node Assumed Natural Strain(ANS) method Drilling degrees of freedom Enhanced Assumed Strain(EAS) method Discrete Shear Gap (DSG) method ….

Partition of unity approximations

44

1 1

1 11 1

( ) ( , )M M

ij i ji j

I f r,s dsdt W f r s

Gauss quadrature

Full integration

( , )u r s urs

0v

xy ur Reduced integration

(Zienkiewicz OC et al., 1971)

r

s

r

s

2 hour–glass modes

Hourglass control (Flanagan & Belytschko, 1981)

Propagation of hour-glass modes through a mesh

Geometric-invariantQuadrilateral element

Comparison with other methods #2 – Reduced Integration method

45

2 24

5 62 21

1 1i ii

r su N u u ua b

2 24

5 62 21

1 1i ii

r sv N v v va b

Incompatible modes (Wilson(1971), Taylor(1976))

Incompatible modes

Only 4-node quadrilateral element Spurious energy in nonlinear analysis

(Sussman T and Bathe KJ, Comput Struct, 2014)

Comparison with other methods #3 – Incompatible modes

46

Distortion sensitivity test


3 3

2 31 1

( ) ( ) ( )i i i i i i i ii i

u h u h x x a h y y a

x

Standard term Additional term

3-node triangular element

Geometric invariant

Pass basic tests

Accurate results than other methods

)( )) ˆ((u x xu u x3

1

( ) i ii

u x h u

3

2 31

ˆ( ) ( ) ( )i i i i i ii

u x h x x a h y y a

Comparison with other methods #4 – Partition of unity approximation

Advantages

47

Proposed remedy to improved membrane response – Partition of unity approximation

12

xx

yy

xy

yz

zx

u ux xv vey y

eu v

ey x

ev w

e z yw ux z

3 4

2 11 1

ˆ( )2

i ii i i i i i

i i

tu h u a f u

V V-∑ ∑3 4

2 11 1

ˆ( )2

i ii i i i i i

i i

tv h v a f v

V V-∑ ∑3 4

2 11 1

( )2

i ii i i i i i

i i

tw h w a f

V V-∑ ∑

3 3

T2 1

1 1

( ), with2

i ii i i i i i

i i

th a h u v w

u u V V u-∑ ∑

Original 3-node displacement interpolation

PU based MITC3+(Enriched MITC3+)

MITC method(transverse shear locking)


High order interpolation(enhanced membrane)

PU approximation

PU based MITC3+ shell element(Enriched MITC3+)

48

Proposed remedy to improved membrane response – Partition of unity approximation

Incremental Green-Lagrange strain tensor components :

Finite rotations :

3 42 2

2 11 1

1 ˆ( , ) ( )2 2

t i t i t ii i i i i i i i n

i i

th r s a f

u u V V V u

3 4

2 11 1

ˆ2

i ii i i i i i

i i

th a f

u u V V u

0 , , , ,1 1( ) ( )2 2

t t t t t t t tij i j i j i j i j i jε g g g g u g g u u u

Kinematics

nonlinear part in the GL strain

Iterative solution procedures

( 1) ( ) ( 1)0 0t i i t t t i K U F

)()1()( iittitt UUU

With the full Newtorn-Raphson iteration scheme, the equation for the i-th iteration in a finite element model are

for the displacement,

Linear

Nonlinear

49

Performances of the PU based MITC3+ element #1

Cantilever beam subjected a tip moment

20, 1, 101200, 0.2

l b h ME v

50


Curved cantilever beam

1 27

4.12, 4.32, 0.1

1 10 , 0.25

R R h

E v

Element Description



MITC3+ 3-node triangular shell elementwith a cubic bubble function (Jeon, 2014)

Enriched MITC3+ PU based MITC3+ (present)


51


Hemisphere shell problem

8

50, 25, 0.25, 90

4.32 10 , 0zL R t f

E v

Element Description



MITC3+ 3-node triangular shell elementwith a cubic bubble function (Jeon, 2014)

Enriched MITC3+ PU based MITC3+ (present)


52

Papers – Nonlinear MITC3+ / Enriched MITC3+ shell elements

The Enriched MITC3+ shell elementwill be submitted in CMAMENonlinear analysis of the MITC3+ shell element

published in Computers and Structures, Jan 2015

53

Future works #1 – PU based finite element method with the FE error estimation

Integrating the PU based FEM with the error estimator

Mesh generation Adaptive mesh refinement algorithm Error estimates Coupling method or indicator

54

4-node partition of unity based finite shell element : Linear dependency problem

Future works #2 – PU based finite element method with reduced integration

2D solid element : No locking treatments due to the high order interpolation

Shell element :Absolutely needs to reduce the transverse shear locking even using the high order interpolation• MITC4-MITC9-MITC16 – Too much computational cost• MITC4-Reduced Integration – Effectively decrease in computational time• Reduced Integration – Reduced Integration – Much faster than others

8-node serendipity element

Hourglass mode

ˆe e e

MITC4Reduced Integration

MITC9 / MITC16Reduced Integration

55

Future works #3 – PU based shell element with high performance computing

“Parallel programming is not a trivial task in most programming languages, and demands a great theoretical knowledge about the hardware architecture and good programming skills” - Neto DP-

Windows Cluster

GPU Cluster

WCU Cluster

Linux Cluster

Partition of unity based shell finite element

( ) ˆu uu x

3

2 31

( ) ( )ˆ i i i i i ii

h x x a h y y au

3

1i i

i

h uu

“The computing of each local matrix is totally independent and so parallelization of these computations is straightforward and can be naturally explored”

56

We proposed two possibilities of using the partition of unity approximation

Increase solution accuracy without local mesh refinements (Part I) Improvement membrane behaviors (part II)

I. PU based MITC3 shell finite element (Enriched MITC3) is reviewed.

This element is obtained by applying linear displacement interpolation covers to the standard 3-node shell element and MITC procedures are used.

The method increases the solution accuracy without any local mesh refinement. We can provide convenience for user by combining PU shell element and FE error

estimates.

II. PU based MITC3+ shell finite element (Enriched MITC3+) is proposed. This shell element is based on the MITC3+ shell element and partition of unity

approximations to improve membrane response. Membrane behaviors of the triangular shell element efficiently is improved by

partition of unity approximation.

Conclusions

57

Conclusion remarks – Most downloaded papers

58

Conclusion remarks – Frontier in the analysis of shells

59

Previous research (2003 ~ 2012)

Virtual heart (NRL project)

Virtual cancer system (NRL project) Angiogenesis and vascular tumor growth

Arterial network with the hemodynamic modelCoronary artery generationK

angw

onN

atio

nal U

nive

rsity

KA

IST,

SB

IE

60

Education & Publications

Education• Ph.D. Student : Korea advanced Institute of Science and Technology (Sep 2010 - present)• MS : Kangwon National University (Sep 2006 – Aug 2008)

(Thesis : Development of a cell-system coupled model of cardiovascular hemodynamics)• BS : Kangwon National University (Mar 2000 – Aug 2006) (Summa Cum Laude & Early Graduation)

International Journals (published : 4)1. Jeon HM, Yoon K, Lee PS. A partition of unity based triangular shell element with improved membrane response, In preparation2. Lee Y, Jeon HM, Lee PS, Bathe KJ. On the behavior of the 3-node MITC triangular shell elements, In preparation.3. Jeon HM, Lee Y, Lee PS, Bathe KJ. The MITC3+ shell element in geometric nonlinear analysis, Computers and Structures, 146, 91-

104, Jan 2015.4. Jeon HM, Lee PS, Bathe KJ. The MITC3 shell finite element enriched by interpolation covers, Computers and Structures, 134, 128-

142, Apr 2014.5. Shim EB, Jun HM, Leem CH, Matusuoka S, Noma A. A new integrated method using a cell-hemodynamics-autonomic nerve control

coupled model of the cardiovascular system. Progress in Biophysics and Molecular Biology, 96(1-3), 44-59, Jan 2008.6. Jun HM, Shim EB. Theoretical analysis of the cross-bridge sliding rate in modulating heart mechanics, International Journal of

Vascular Biomedical Engineering, 5(2), 34-45, Oct 2007.

Presentation (27)• Jeon HM, Yoon K, Lee PS. Development of the enriched MITC3 shell element. KSME Annual Conference, 192-193, Apr 2014.• …..• Computational study on the arterial tree generation based on blood volume optimization. KSME Annual Conference, Oct 2005.

Experience• URP program, Kangon National University (Sep. 2003 ~ Aug 2006), Dep. of Mechanical Engineering• Visiting Researcher, Kyoto University (Dec 2007- Mar 2008 and Apr 2009- Jun 2009), Graduate School of Medicine• Visiting Researcher, MIT (Aug 2008 – Oct 2008), Dep. of Health Sciences and Technology • Visiting Researcher, Oxford (Jul 2010- Aug 2010), Centre for Mathematical Biology• Research Assistant, KAIST (Jan 2010 – Aug 2012), Dep. of Bio and Brain Engineering, Advisor : Prof. Kwang-Hyun Cho.

Documents

Partition of Unity Based Shell Finite Elements