BEM research and new developments in BrazilBEM research a tiny part of and new developments in Brazil Ney Augusto DumontNey Augusto Dumont Civil Engineering Department – PUC-Rio

BEM researchand new developments in Brazil

Ney Augusto DumontNey Augusto DumontCivil Engineering Department – PUC-Rio

NSF_BEM_workshop: BEM research and new developments in Brazil

BEM research a tiny part of

and new developments in Brazil

Ney Augusto DumontNey Augusto DumontCivil Engineering Department – PUC-Rio

BEM26 – CONVENTIONAL, HYBRID AND SIMPLIFIED BOUNDARY ELEMENT METHODSNSF_BEM_workshop: BEM research and new developments in Brazil

(A view from this tiny part of Brazil: my office room’s window)


Brazil: my office room s window)

Prominent BEM-researchers in Brazil• José Cláudio Faria Telles• José Cláudio Faria Telles• Webe João Mansur• Luiz Wrobel (at Brunel University)

And large team at COPPE/UFRJ andother universities

Luiz Wrobel (at Brunel University)

• Euclides de Mesquita NetoA UNICAMP

• Paulo SolleroAt UNICAMP

• Wilson Sérgio Venturini (passed away last July)

• João Batista de PaivaHumberto Breves Coda

At São Carlos (UESC)

• Humberto Breves Coda

• Delfim Soares Júnior – at UFJF New generationDelfim Soares Júnior at UFJF• Francisco Célio de Araújo – at UFOP


g

ANNOUNCEMENT• BETEQ 2011 - XII International Conference on

Boundary Element and Meshless Techniques12 15 J l 2011 B ili B il12-15 July 2011, Brasilia, Brazil A few days in Rio de Janeiro is mandatory!


ANNOUNCEMENT

A few days in Rio de Janeiro is mandatory!


R tl l d d Ph D Th i

• Recently graduated student and works in progress at PUC-Rio

Recently concluded Ph.D Thesis:• M. F. F. Oliveira, Conventional and simplied-hybrid boundary element

methods applied to axisymmetric elasticity problems in fullspace and halfspace, PUC-Rio (2009). Co-advisor: Patrick Selvadurai (McGillhalfspace, PUC Rio (2009). Co advisor: Patrick Selvadurai (McGill University)

Ph.D Theses in progressp g• C. A. Aguilar M., Comparison of the computational performance of the

advanced mode superposition technique with techniques that use numerical Laplace transforms (since September 2008).

• D. Huamán M., Gradient elasticity formulations with the hybridboundary element method (since September 2008)

M S Th i iM.Sc. Thesis in progress• E. Y. Mamani V., Application of a generalized Westergaard stress

function for the analysis of fracture mechanics problems (since January2010) Co-advisor: A A O Lopes2010). Co-advisor: A. A. O. Lopes


(Intended) Contents of this presentation• Variationally-based, hybrid boundary and finite elements

• Developments for time-dependent problems

Developments in gradient elasticity• Developments in gradient elasticity

• Dislocation-based formulations (for fracture mechanics)Dislocation based formulations (for fracture mechanics)

• From the collocation (conventional of hybrid) boundary ( y ) yelement method to a meshless formulationOr: The expedite boundary element method


Approximations on the boundary

0, =+ ijji bσ 0,

=+ ip b

jjiσ

mimi duu =

ttt

pim

pmi

duu =

ttt ii = pi

p ttti=

pm m

dd , : nodal atributesptt , : surface atributes

• NSF_BEM_workshop: BEM research and new developments in Brazil

Fundamental solution

0** =Δ+ mimjji pσ

* * * * * * in r ri im m i im m is sm mu u p c u p u C p= + ≡ + Ω

, mimjji p

Ω== in **,

***mpkmijkpmijmij puDpσσ

Γ== on *****mjjimmimi pptt ησ

• Properties

mjjimmimi pp η

imjjim d δσ −=Ω∫Ω *, imim δdt −=Γ∫Γ

*


The conventional, collocation BEM

)()( pp ttGddH −≅−Derived from the Somigliana’s identity

≅ means congruence in terms of weighted residuals (collocation method)

pp td , are already an expedite means of taking a particular solution into account

(inherent approximation error due to rigid body displacements!)

, are already an expedite means of taking a particular solution into account

or ttutGdHdu iimmnmnninjjim ⎟⎠⎞⎜

⎝⎛ Γ≡≅≡⎟

⎠⎞⎜

⎝⎛ Γ ∫∫ Γ

∗

Γ

∗ ddησ⎠⎝⎠⎝ ∫∫ ΓΓ

Where:functionsion interpolatnt displaceme

attributesnt displaceme nodal==

in

n

ud

Where:

functionsion interpolat forcetraction attributes forceraction boundary t

==

i

i

tt

∗∗ ∗∗imjim u,σ Stress and displacement expressions of the problem’s fundamental solution


Linear algebra properties of H

1=∗pΩ IHH =+

Γ

Ω

Γη

pp

ηΓ

Spectral properties:

PACAM 2010 – Foz do Iguaçu, Brazil Spectral properties of the double layer potential matrix H

Consistent formulation of the BEM

( ) ( )p pR⊥− = −H d d GP t t( ) ( )

Where the orthogonal projector

∫

( ) 1T TR R

−⊥ = − = −P I P I R R R R

∫Γ Γ= drisis utRwith

f h f h h i bi f i id b d di l i h

* * * * * * in r ri im m i im m is sm mu u p c u p u C p= + ≡ + Ω

comes from the fact that there is an arbitrary amount of rigid body displacements in the fundamental solution


Literature Review – part IHybrid formulations: Coined by Pian in 1967 “... to signify elements which maintain either equilibrium or compatibility in the

element and then to satisfy compatibility or equilibrium respectively along the interelement boundary”

Parallel developments

Trefftz E 1926 Ein Gegenstück zum

Hellinger, E., 1914. Die allgemeinen Ansätze der Mechanik der Kontinua, Enz. math. Wis, 4, 602-694.

Trefftz, E., 1926. Ein Gegenstück zum Ritzschen Verfahren. Proc. 2nd International Congress of Applied Mechanics, Zurich, Switzerland.

Reissner, E. 1950. On a variational theorem in elasticity, J. Math. Phys., 29, 90-95 .

Hu, H.-C., 1955. On some variational principles in the theory of elasticity and the theory of plasticity, Scientia Sinica, 4, 33-54

Pian, T. H. H., 1964. Derivation of element stiffness matricesby assumed stress distribution. AIAA J., 2, 1333-1336by assumed stress distribution. AIAA J., 2, 1333 1336

Jirousek, J. & Leon, N., 1977. A powerful finite element forplate bending, Com. Meths. in Appl. Mech. Engng., 12, 77-96.

Qin Q H 2003 The Trefftz Finite and Boundary ElementDumont, N. A., 1989. The hybrid boundary element method: an alliance between mechanical consistency and simplicity. Qin, Q. H., 2003. The Trefftz Finite and Boundary Element

Method, WITPress.

(Many variations)

y p yApplied Mechanics Reviews, 42, no. 11, Part 2, S54-S63

(and variations)

Present theoretical investigationMotivation: we are here

Double layer potential matrix

Mechanical assumptions,

Theoretical tool:

p ,linear algebra consequences

Displacement virtual work principle (elastostatics): The Ouroboros (from Wikipedia)

PACAM 2010 – Foz do Iguaçu, Brazil Spectral properties of the double layer potential matrix H

The Ouroboros!

Collocation

BEM(Tail?)

Variational issuesIntegration issues

Hybrid

BEM Expedite

BEM(Head?)

( )

Integration issuesFundamental solutions

Matrix spectral properties(Generalized inverses)

Consistency issues

HybridDisplacement

BEM ConsistentCollocation

Applications

Galerkin SimplifiedH b id

BEM(Here and

now)

(from Wikipedia)BEM

Hybrid

BEMSymmetricGalerkin

BEM

BEM/MRM 32 – The boundary element method revisited

Dumont, N. A.: "The hybrid boundary element method – fundamentals", in preparation tobe submitted to Engineering Analysis with Boundary Elements

The conventional, collocation BEM


The hybrid BEM

Dumont N A : “Variationally-Based Hybrid Boundary Element Methods”


Dumont, N. A.: Variationally-Based, Hybrid Boundary Element Methods ,Computer Assisted Mechanics and Engineering Sciences (CAMES) Vol 10 pp 407-430, 2003

The hybrid displacement BEM


Definition of the matrices involved

: d

: d

mn jim j inH u

G u t

σ η∗

Γ

∗

= Γ

Γ

∫∫

1=∗pΩ

pp: d

: d

m im i

mn jim j in

G u t

F uσ η

Γ

∗ ∗ ∗

Γ

= Γ

= Γ

∫∫ η

Γ

Ω

Γη

p

: dm im iL u t

Γ

Γ= Γ

∫∫

η

Simplified HBEM: ∗ ∗←F HU


p

The Ouroboros!

External reference system( ),d p

Some isolated virtual work statements

T ∗ ∗= ⇔ =H p p Hd dT L∗ ∗⇔Gt d G d

( ),∗ ∗p d Internal reference system

T L= ⇔ =Gt d G p dT L= ⇔ =L t p Ld d

( ), Lt d System of Lagrange multipliers

= ⇔ =L t p Ld dThere are more virtual work statements;Several restrictions apply!

G and L should be rectangular, in general;G should be consistent

Dumont, N.A., An assessment of the spectral properties of the matrix G used in the boundary element methods Computationalmatrix G used in the boundary element methods, Computational Mechanics, 22(1), pp. 32-41, 1998


The Ouroboros!

External reference system( ),d p

Some isolated virtual work statements

T ∗ ∗= ⇔ =H p p Hd dT L∗ ∗⇔Gt d G d

( ),∗ ∗p d Internal reference system

T L= ⇔ =Gt d G p dT L= ⇔ =L t p Ld d

( ), Lt d System of Lagrange multipliers

= ⇔ =L t p Ld dThere are more virtual work statements;Several restrictions apply!

G and L should be rectangular, in general;G should be consistent

∗ ∗ ∗=F p d =Kd pResults at internal points: no integral statement actually required


Spectral properties for the hybrid BEM

For a finite domain:

W W⊥ = −P I PWP is the orthogonal projector onto the

space of rigid body displacements and

TW V= ⇒ =HP 0 H P 0Then,

W WW

Definition ofVP

V∗ =F P 0

KdFor consistency,

( ) 1T −∗K H F P H =Kd p

Results at internal points: no integral statement actually required!

and ( )TV

∗= +K H F P H in

Results at internal points: no integral statement actually required!

∗ ∗ =F p Hd∗p solved from either or T ∗ =H p p with V∗ =P p 0

and* * * ri im m iu u p c= + * * * ij ijm mpσ σ=


Some applications already implemented• 2D and 3D potential and elasticity problems• 2D and 3D acoustics• 2D general time-dependent problems (in the

frequency domain):• Advanced mode superposition analysisAdvanced mode superposition analysis• Use of numerical inverse transforms

• 2D sensitivity analysis• 2D FGM-analysis for potential problems• 2D fracture-mechanics analysis – evaluation of stress

intensity factorsintensity factors• Axysimmetric fullspace and halfspace elastostatics• 2D (and 3D) gradient elasticity analysis( ) g y y


Example of application of non-singular fundamental solutions: two dimensional transient heat conduction in a homogeneous square plate

y

)1,1()1,0(

1)( =tu

0)( =tu 1)( =tu0)0,,( =yxu

x)0,0( )0,1(0)( =tu

1k Conductivity Distorced 4x4 mesh1=k

1=c

Conductivity

Specific heat

Distorced 4x4 mesh

Temperature results along the edge x = 0 for several time instants

Analytic solution

Classical modal analysisClassical modal analysis.

Serendipity elements.Quadratic elements.

Hybrid formulation3 mass matrices.

Westergaard’s Stress Function as a Fundamental Solution

- Example 8 – inclined edge crack in a rectangular plate

3.50

4.50

ctor

15

1.50

2.50

Stre

ss In

tens

ity F

a

KI - NumericKII - NumericKI - AnalyticKII - Analytic

10 σ=1

π83

15

π83

-0.50

0.50

3 4 5 6 7 8 9 10 11 12 13 14 15

S 25

-1.50

Number of Elements Along the Crack


1,5

Example 3 – semicircular crack in an infinite continuum

1,1

1,3

0,7

0,9

ity fa

ctor

s

KI - Ratio 1.0

KII - Ratio 1.0

0,3

0,5

Stre

ss in

tens KI - Analytical

KII - Analytical

KI - Ratio 1.06

KII - Ratio 1.06

σ=1 2

0 3

-0,1

0,1

0 10 20 30 40 50 60 70 80 90 100Number of elements

S

-0,5

-0,3


34

Modeling a curved crack

25

2

34

5

y

1 6

0

1

a

x

y

2

3

4

5

1

2 5

Previous development: superposition of elipsis

Current development:Current development:Superposition ofhalfcracks

Inspired by:p yTada, H., Ernst, H. A., Paris, P. C. (1993), Westergaard stress functions for displacement-prescribed crack problems – I, Int. Journal of Fracture, Vol. 61, pp 39-53


26

F

36 45

47

49A

B

9

14

1


‐0.44

‐0.43

U potential along the line segment A‐B due to an external source

26

‐0.46

‐0.45

36 4547

49

F

A

‐0.48

‐0.47

U poten

tial

9

14

B

‐0.5

‐0.49

1

‐0.52

‐0.51

0 5 10 15 20 25 30 35 40 45 50

Analityc

numeric

Number of point along the line dashed A‐B


0.010

qx gradient along the line segment A‐B due to an external source26

F

0.009

9

36 4547

49

F

A

B

0 007

0.008

qx gradien

t

9

14

0.006

0.0071

0.005

0 5 10 15 20 25 30 35 40 45 50

Analityc

numeric

0 5 10 15 20 25 30 35 40 45 50

Number of point along the line dashed A‐B


1 00

1.05

1.10

alyt

ic

Stress intensity factors in terms of p* and J integral - node A

0.90

0.95

1.00

umer

ic /

KI a

na

Analytic

from p*1 1.0 0.51.0

q=1

a1 b

p*1

0.2 a2

p*2

a3 A

0.80

0.85

0 20 40 60 80 100

KI n

Number of elements along the crack

from p 1

from JA B

0.2

a

g


1 00

1.05

1.10

lytic

Stress intensity factors in terms of p* and J integral - node B

0.90

0.95

1.00

umer

ic /

KI a

nal

Analytic

from p*n 1.0 0.51.0

q=1

an b

p*np*n‐1

an‐1 0.2Ban‐2

0.80

0.85

0 20 40 60 80 100

KI n

u

Number of elements along the crack

from p n

from JA B

a

0.2

g


1.5

Gradient qy along the line segment y=0.20

0 9

1.1

1.3

1.5

y

Analytic

q=1

0.5

0.7

0.9

Gradien

t qy 3 Elements

7 Elements

19 Elements

49 Elements

q 1

2

4 0.20

‐0.1

0.1

0.349 Elements

99 Elements

2

‐2 ‐1.5 ‐1 ‐0.5 0 0.5 1 1.5 2

Points along the dashed line

EXAMPLE: gradient qz along a line segment

0.25

0.20

Analytic

48 elements

0.15 76 elements

0.10

q z

0.05

0.00

-0.050 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

R. A. P. Chaves, “The Hybrid Simplified Boundary Element Method Applied to Time-Dependent Problems”, Ph.D. Thesis, PUC-Rio, Brazil, 2003

Numerical example

Non-convex axisymmetric volume with multiply connected surfaces

Unitary ring sources (10, -5)

BETEQ 2009 - International Conference on Boundary Element Techniques Athens, July 2009

An expedite formulation of the BEMSome numerical assessments for 2D potential problems

( )π4

)()(ln),(:sol_92

12

1 yyxxyxu −+−=

Numerical and analytical values of potentials at internal pointsof potentials at internal points

T ∗ =H p pT ∗ =V p 0

* * * r+ ri im m iu u p c= +

ICCES 2010 – AN EXPEDITE FORMULATION OF THE BOUNDARY ELEMENT METHOD

An expedite formulation of the BEMSome numerical assessments for 2D potential problems

( )π4

)()(ln),(:sol_92

12

1 yyxxyxu −+−=

* * *ij ijm mpσ σ=

Numerical and analytical values of gradients at internal points


Contents of this presentation• Variationally-based, hybrid boundary and finite elements




• From the collocation (conventional of hybrid) boundary ( y ) yelement method to a meshless formulationOr: The expedite boundary element method


GENERALIZED HELLINGER-REISSNER POTENTIAL

( )( )∫(∫ ( )∫ ~~1 +Γ−+Ω−−− duuduuut ηδσρδδσ( )( )∫(∫ ( )∫( )∫ ( )∫ ) 0~,~

,0

=Γ−−Ω−++

+Γ+Ω

ΓΩ

ΓΩ

dtdtudufu

duuduuu

ijijiiijiji

iijijt iiijij

ησδρσδ

ηδσρδδσ

( ) ( ) )ΓΩ jjjj

0~δu Γalong0=δ iu uΓalong

δui = 0 at both time interval extremities t0 and t1i

Literature Review – part I (continued)

Przemieniecki, J.S. (1968). Theory of Matrix Structural Analysis, Dover Publications, New York (displacement-based free vibration analysis – truss and beam):

Voss, H., 1987, A New Justification of Finite Dynamic Element

( ) ( ) )( 864

642

40

20 ωωωω O+−−−−−= KMKMMKK

Gupta, Paz: several other displacement-based elementsVoss, H., 1987, A New Justification of Finite Dynamic Element Methods, Numerical Treatment of Eigenvalue Problems, Vol. 4, 232-242, eds. J. Albrecht, L. Collatz, W. Velte, W. Wunderlich, Int. Series on Num. Maths. 83, Birkhäuser Verlag, Stuttgart

Gupta, Paz: several other displacement based elements for plane-state problems. Coined finite dynamic elements, although they only have been applied to free vibration analysis (better, then: finite harmonic elements!)

?

( ) ( ) ( ) )( 864

642

420

20 ωωωω O+−−−−−−= KMKMKMKK

?

Dumont N A & Oliveira R (1993) 1997 The Dumont N A & Oliveira R 2001 From frequency-

Directly based on Pian and Przemieniecki: General , consistent finite and boundary dynamicelement families: acoustics, free vibration, transient problems via an advanced modal analysis.

Dumont, N. A. & Oliveira, R., (1993) 1997. The exact dynamic formulation of the hybrid boundary element method, Procs. XVIII CILAMCE, Brasília, 29 - 31 October, 357-364

Dumont, N. A. & Oliveira, R., 2001. From frequency-dependent mass and stiffness matrices to the dynamic response of elastic systems. Int. J. Sol. Struct., 38, 10-13, 1813-1830

Literature Review – part IILancaster, P., 2002. Lambda-Matrices & Vibrating Systems, Dover Publications(from 1966)

d...)HHH(p...)FFF( 24

12

0*

24

12

0 +ω+ω+=+ω+ω+

pp)HHH( *T4T2T +ω+ω+ pp...)HHH( 210 =+ω+ω+Dumont, N. A.: “On the Inverse of Generalized Lambda Matrices with Singular Leading Term”, IJNME, Vol 66(4), 571-603, 2006

)(pd)MMK( 42 ω+ωωH F H KT − =10VH T

0 = 0VF 0 =

)(pd...)MMK( 24

12

0 ω=+ω−ω−(Przemieniecki)

Dumont, N. A.: “On the solution of generalized non-linear complex-symmetric eigenvalue problems”, IJNME, Vol 71, 1534-1568, 2007

Sleipjen, G. L. G., Van der Horst, H. A., 1996, A Jacobi-Davidson iteration method for linear eigenvalue problems,

Nonlinear eigenvalue problem:

SIAM J. Matrix Anal. Appl., 17, 401-425Arbenz, P, Hochstenbach, M. E., 2004. A Jacobi-Davidson method for solving complex-symmetric eigenvalue

problems, SIAM J. Sci. Comput. 25, 5, 1655-1673

Problems with viscous damping

( ) 0ΦΩMΦΩCΦK =+−∑=

−n

j

jj

jji

1

2120

0000K ⎤⎡ 0

Nonlinear eigenproblem:

0CM00CMC0

MMCM00000K

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

ΦΦΦ

ΦΦΦΦΦΦ

⎥⎥⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎢⎢⎡

−

−

1,11110

1,00100

32

322

221

0

n

nn

iii

i …Augmented formulation:

000CMCMMMCMC

000M0

⎤⎡Ω⎥⎤

⎢⎡ ΦΦΦ⎥⎥⎤

⎢⎢⎡

⎥⎦

⎢⎣ ΦΦΦ

⎥⎥⎥

⎦⎢⎢⎢

⎣

−

−−−−

01001003221

2211

1,11,10,1

n

n

nnnn

n

iiii

……

0

00

00

0000M

0CM0CMC

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

Ω

Ω

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

ΦΦΦ

ΦΦΦ

⎥⎥⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢⎢⎢

⎣

−

−−−−−

−

1

1

0

1,11,10,1

1,11110

1,00100

32

322

3221

nnnnn

n

n

iii …

0000M ⎥⎦⎢⎣ n

ΦΩΦΩΦΩΦΩ

⎞⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛+∑ ∑∑

= =

−−

=

−−

jj

n

j

j

k

kjj

kj

k

kjj

k i

1222

1

2

1

212

2

22 IMC TT

Orthogonalityproperties:

ΩΦΩΦΩΦΩΦΩΦΦ =⎟⎟⎠

⎞⎜⎜⎝

⎛++∑ ∑∑

=

−

=

−−

=

−−n

j

j

k

kjj

kj

k

kjj

k i1

12

1

222

1

120 MCK TTT

properties:

Summary

d2∂ in

K d M d p0

2

21

1− −∂∂

==∑ ( ) ( )i

i

i

ii

n

tt

d d d2 4 6∂ ∂ ∂

(4)Equation in the time domain

K d M d M d M d p0 1

2

2 2

4

4 3

6

6+∂∂

−∂∂

+∂∂

=t t t

t( )(Set of coupled, higher-order differential equations)

e. g.

d ΦMode superposition : (15)

( ) 0MK =− ϕωωω )()( 2from

T2 ΦΩ

d = ΦηMode superposition : (15)

pT2 Φ=+Ω ηη (17)

(Set of uncoupled, second-order differential equations) ( p , q )

Frequency-domain formulation

0, 2 =++ iijji ukb ρσ ζωω ik 222 +=where

(Laplace/Fourier transforms or)Solution:

St t l d i

(Laplace/Fourier transforms or)

Advanced mode-superposition techniqueSolution:

( ) ( )bbb ppT −=−+− ΦηηηηΩ2

Structural dynamics

( ) ( )bbb ppT=+ ΦηηηηΩ

Diffusion-type problems

( ) ( )pp −=−+− ΦηηηηΩ

Structural dynamics with viscous damping

( ) ( ) ( )bbb i ppT −=−−− ΦηηηηΩ

Initial displacements and velocities

[ ] dKK TT00 elelelel ΦΦΦη

1−= [ ] 00 elelelelη

dMT1rigrig Φη =

In case of viscous damping:

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

2

1

2211

21

ηη

ΩΦΩΦΦΦ

ddi ⎭⎩⎦⎣⎭⎩ 22211 η

⎬⎫

⎨⎧

=⎬⎫

⎨⎧⎥⎤

⎢⎡ dKKK TTT

011201101 ΦηΦΦΦΦ

⎭⎬

⎩⎨

⎭⎬

⎩⎨⎥⎦

⎢⎣ dKΩKΩK TTT

01222011101 ΦηΦΦΦΦ i

EVALUATION OF RESULTS AT INTERNAL POINTS

[ ])()()()( ω−ωω=ω∗ bddSp in which )()()( 1 ωωω HFS −=

l d breplaced by

[ ]∑∗n

bi2 )()()( ddS ∑∑∑−

ω⎟⎞

⎜⎛

ω=ωn

in

in

i 21

22 HFSwith[ ]∑=

ω−ωω≈ωi

bi

i

0

2 )()()( ddSp ∑∑∑===

ω⎟⎠

⎜⎝

ω=ωi

ii

ii

i000

HFSwith

( )∑∗n

bi2)( ΦΩS ( )∑=

∗ −=i

biit

0

2)( ηηΦΩSp

EVALUATION OF RESULTS AT INTERNAL POINTS

( )n nm n

( )∑=

∗ −=n

i

biit

0

2)( ηηΦΩSp ∑∑∑=

∗∗

= =

∗∗ ≡ω=i

ii

j ii

ijt

0

2

0 0

2)( pΩupuu

( )∑∑ ∗ −=n i

bit 2)( ηηΦΩSuu ( )∑∑= =

=i j

j-ijt0 0

)( ηηΦΩSuufor n = 3

( ) ( )[ ΦΩSuSuSuΦΩSuSuΦSuu 4021120

2011000)( ∗∗∗∗∗∗ ++++++=t ( ) ( )[

( ) ] ηΦΩSuSuSuSu

ΦΩSuSuSuΦΩSuSuΦSuu6

03122130

021120011000)(∗∗∗∗ ++++

++++++t

p F Hd Sd∗ −= ≡1S is required for

Example 1: Initial velocity – Displacement at point A

0.0045

0.0025

10 m

2 5 /(10 3)

E = 210,000 Mpa ρ = 7,850 kg/m3

0.0005 A

30 m

v0 = -2.5 m/s(10, 3)

Analytic

34 elements

-0.0015

-0.0035

-0.00550 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Transient Analysis – Example 2: Gravitational force

59 ll d59 equally spaced,linear boundary elements

g = -9.81 m/s2

35 m

00 M

pa

g 9.81 m/s

m

(20, 30)E =

210,

00

0 kg

/m3 ( ) ( )[ ]

( )∑∞

= ωπ−ω−

=1

212cos1ksing4u

n n

nn

ntx

50 m

(0 15)

ρ =

7,85

ρ=ω Ek nn

(25, 10)

(0, 15)( )

L212k π−

=n

n

(35, 0)(15, 0)

υ= 0 (can be solved in the frame of the theory of potential)

0.00002Example 2: Gravitational force – Displacement at point A

-0.00008a

-0.00018

g = -9.81 m/s2

35 m

(20, 30)E =

210,

000

Mpa

0 kg

/m3

0 0002850

m(25, 10)

(0, 15) (10, 15)A

E

ρ =

7,85

0

Analytic-0.00028 ( , )

(35, 0)(15, 0)59 elements

-0.00038

-0.000480 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

-0.00001Example 2: Gravitational force – Displacements along a line segment

t 0 004

-0.00011

t=0.004

-0.00021 t=0.008

-0 00041

-0.00031

9 81 / 2

35 m

00 M

pa

t=0.012

-0.00051

-0.00041 g = -9.81 m/s2

50 m

(20, 30)

(20, 26.2)

E =

210,

00

= 7,

850

kg/m

3

-0.00061 (25, 10)

(0, 15)

ρ =

t=0.016Analytic

59 elements

-0.000710 5 10 15 20 25

(20, 1) (35, 0)(15, 0) t=0.0259 elements

Plane frame with 6 members and 12 d.o.f. b itt d t t i l lsubmitted to a triangular pulse

P(t)

5

2 P(t)P(t)4

6

1

2

3

1

3

1

5

L

P(t)

P1

2P(t)8

7

9

11

1210

3 42

33L1t 2t0 t

12

5 6

y

x

64

3L(Weaver, W., Jr., and Johnston, P. R., 1987. Structural Dynamics by Finite Elements, Prentice-Hall Inc., Englewood Cliffs, New Jersey )

zL L 3L L 2L


Plane frame with 6 members and 12 d.o.f. b itt d t t i l lsubmitted to a triangular pulse

Displacement results with 1 and 4 mass matrices, as compared with values given by Weaver and Johnston.

Eigenvalues obtained for the problem with 1, 2, 3 and 4 mass matrices.






• From the collocation (conventional or hybrid) boundary ( y ) yelement method to a meshless formulationOr: The expedite boundary element method


A Family of 2D and 3D Hybrid Finite Elements for Strain Gradient Elasticity

Ney Augusto DumontDaniel Huamán Mosqueirae-mail: [email protected]

BETeq – International Conference on Boundary Element & Meshless Techniques XI12-14 July 2010, Berlin

Conclusions (1)

kkjijiji g ,2ττσ −= Total stress

• The hybrid BEM / FEM (a two-field formulation) is a natural variational tool to deal withgradient elasticity

• Singular and nonsingular fundamental solutions were either redeveloped for BEM or originallydeveloped for FEM gradient elasticity implementations

• General families of finite elements were obtained

• Extension to time-dependent problems in the frequency domain is straightforwardExtension to time dependent problems in the frequency domain is straightforward(in progress – already done for truss and beam elements)

• Simple implementations for truss and beam elements: was a fruitful apprenticeship(disagreement with some results in the literature - symmetry and representation of constant( g y y pstrain state)

• Treatment of the normal displacement gradient on Г is different from Mindlin’sproposition (our results still remain to be validated)

• Numerical examples for 2D problems are being implemented


Conclusions (2)

kkjijiji g ,2ττσ −= Total stress

• .• .

jjj ,

• .• Treatment of the normal displacement gradient on Г is different from Mindlin’s

proposition (our results still remain to be validated)

• Numerical examples for 2D problems are being implemented:

• Orthogonality to rigid body displacements: OK!

• Symmetry of the flexibility matrix F*: OK!• Symmetry of the flexibility matrix F : OK!

• Patch tests for linear displacements fields: OK!

• Patch tests for non-linear displacement fields: convergence still questionable.






• From the collocation (conventional or hybrid) boundary ( y ) yelement method to a meshless formulationOr: The expedite boundary element method


Conclusions

• Implementation extremely advantageous for:

Expedite formulation of the boundary element method (in progress)

• large problems • problems with complicated fundamental solutions (time-dependent, axisymmetric, for gradient elasticity, etc)

• Numerical accuracy still under investigation, but comparable to theNumerical accuracy still under investigation, but comparable to the conventional BEM•Evaluation of results at internal points requires no further integrations•However, results close to the boundary require the knowledge of the null

V ( hi h b b i d i G S id l i i i i i lspace V (which may be obtained via Gauss-Seidel iteration in principle with a simple pre-conditioning of HT)

Work in progressWork in progress•Implementation in Fortran for large 2D potential and elasticity problems in the frequency domain•Implementation for strain gradient elasticityImplementation for strain gradient elasticity


Documents

BEM research and new developments in BrazilBEM research a tiny part of and new developments in Brazil Ney Augusto DumontNey Augusto Dumont Civil Engineering Department – PUC-Rio