Finite element method: the basics

Finite element method: the basicsProf. José E. Andrade

Department of Civil & Environmental Engineering

July, 2007

PLAXIS FINITE ELEMENT CODE FOR SOIL AND ROCK ANALYSES

Plaxis BVPO Box 572

2600 AN DelftThe Netherlands

Tel: +31 (0)15 251 77 20Fax: +31 (0)15 257 31 07

E-Mail: [email protected]: www.plaxis.nl

19 - 21 March 2007 Course Computational Geotechnics (German)Stuttgart, Germany

26 - 29 March 2007International course for experienced Plaxis usersAntwerpen, Belgium

1 - 4 April5th International Workshop on Applicationsof Computational Mechanics in Geotechnical EngineeringGuimaraes, Portugal

18 - 19 April 2007Plaxis user meeting UKLondon, Manchester, United Kingdom

25 - 27 April 2007NUMOG XTenth International Symposium on Numerical Models in GeomechanicsRhodes, Greece

5 - 10 May 2007ITA-Aites World Tunnel Congress 2007Prague, Czech Republic

8 - 11 May 200716th Southeast Asian Geotechnical ConferenceSelangor Darul Ehsan, Malaysia

12 - 14 June 2007 Course Computational GeotechnicsManchester, United Kingdom

25 - 28 June 2007Course on advanced Computational GeotechnicsSydney, Australia

16 - 20 July 2007XIII PCSMGE 2007Isla de Margarita, Venezuela

24 - 27 July 2007Course on Computational Geotechnics and DynamicsChicago, USA

10 - 14 September 2007Course on advanced Computational GeotechnicsAnkara, Turkey

24 - 27 September 2007XIV ECSMGE Madrid, Spain

21 - 24 October 200710 ANZ SMGE,Brisbane, Australia

7 - 9 November 200714th European Plaxis User MeetingKarlsruhe, Germany

26 - 28 November 200714 African Regional Conference SMGE, Yaounde, Cameroon-Africa

10 - 14 December 200713th Asian Regional Conference on Soil Me-chanics and Geotechnical EngineeringCalcutta, India

Activities 2007

7005

910

Outline

• Applications of FEM

• Fundamentals

• Examples

Applications of FEM

Biomechanics

bypass alternatives

patient-specificmodel

Geomechanics

COMPACTIVEZONE

DILATIVEZONE

!a

r

GRAIN

VOID

SHEARBAND

P

FIELD SCALE SPECIMEN SCALE

'HOMOGENEOUS'

SOIL

MESO SCALE GRAIN SCALE

-3 -2 -1 0 >1LOG (m)

FOOTING

FAILURE

SURFACE!

Solid-fluid interactions

DEVIATORIC STRAIN

0 .0 5

0 .1

0 .1 5

0 .2

0 .2 5

0 .3

shear strain

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

4

PRESSURE, kPa

VERT

ICA

L C

OO

RDIN

ATE

, m

FE SOLNANAL SOLN

consolidation

Simulation engineering

Fundamentals of FEM

FEM

• Designed to approximately solve PDE’s

• PDE’s model physical phenomena

• Three types of PDE’s:

• Parabolic: fluid flow

• Hyperbolic: wave eqn

• Elliptic: elastostatics

!!"!#$

%!"!#$

&

&

' '() '(* '(+ '(, '(- '(. '(/ '(0 '(1 )'

'()

'(*

'(+

'(,

'(-

'(.

'(/

'(0

'(1

)

'

'()

'(*

'(+

'(,

'(-

'(.

'(/

'(0

'(1

)

FEM recipe

Strong from

Weak form

Galerkin form

Matrix form

Elastostatics: strong

PDE

B.C.’s

u,xx +f = 0u(1) = g

u,x (0) = h

strong form

derived from continuum mechanicsstrong form = PDE + B.C.’s

usually, there is no exact sln for strong form

x10

Elastostatics: weak

•Use principle of virtual work•Introduce virtual displacement•Use strong form∫ 1

0

w(u,xx +f) dx = 0

w; w(1) = 0

∫ 1

0

w,x u,x dx =

∫ 1

0

wf dx + w(0)h

Elastostatics: Galerkin

Construct approximate solution uh

= vh

+ gh

like w

like g

Construct functions based on shape functions

wh

=

n∑

A=1

NAcA vh

=

n∑

A=1

NAdA

gh= gNn+1

Piecewise linear FE

x

10

1

N1

Nn+1

NA

xn+1

xn

x1

x2

xA-1

xAxA+1

finite element

node

Elastostatics: matrixUse weak form, plug-in Galerkin approximation

n∑

B=1

KABdB = FA KAB =

∫ 1

0

NA,xNb,x dx

FA =

∫ 1

0

NAf dx + NA(0)h −

∫ 1

0

NA,xNn+1,x dxg

it all boils down to...

stiffness matrix

K · d = F

force vector

Properties of K · d = F

•Stiffness matrix is•symmetric•banded•positive-definite

•Displacement vector = unknowns only•Can use any linear algebra solver to find solution

may not alwaysapply

Multi-D deformation

!g

!h

!

∇ · σ + f = 0 in !

u = g on "g

σ · n = h on "h

equilibriume.g., clampe.g., confinement

Constitutive relation ugiven get !

e.g., elasticity, plasticity

FEM programTIME STEP LOOP

ITERATION LOOP

ASSEMBLE FORCE VECTOR

AND STIFFNESS MATRIX

ELEMENT LOOP: N=1, NUMEL

GAUSS INTEGRATION LOOP: L=1, NINT

CALL MATERIAL SUBROUTINE

CONTINUE

CONTINUE

CONTINUE

T = T + !T

Element technology: 2D

Serendipity family of quads

Lagrange family of quads

Standard triangularelements

displacement nodeGauss integration point

Modeling ingredients

1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve

4.417

B

H










!a

!r




!a

!r

Examples

Hyperbolic: LSST array

output

input

Geometry and B.C.s

input

output

Material parameters

shear modulus degradation & damping

Input base acceleration

Acceleration output

Soil-fluid interaction

• Nonlinear continuum mechanics

• Robust constitutive theory

• Computational inelasticity

• Nonlinear finite elements

Model ingredients

!s

"#

"

x1

x2

X

x

!f

"#

f






Model ingredients

! =’"

! !’ = ’2 3






Model ingredients

!"

!#

!$

!n

!n+1

!n+1

tr

Fn+1

Fn






Model ingredients

Pressure nodeDisplacement node

Plane-strain compressspecific volume

CT scan FE model

Plane-strain compressspecific volume

shear strain and flow fluid pressure

Plane-strain compress

Questions?

Documents

Finite element method: the basics