5th International Workshop on Applications of Computational Mechanics in Geotechnical Engineering Guimaraes, Portugal 18 - 19 April 2007 Plaxis user meeting UK London, Manchester, United Kingdom 25 - 27 April 2007 NUMOG X Tenth International Symposium on Numerical Models in Geomechanics Rhodes, Greece

Course on advanced Computational Geotechnics Ankara, Turkey 24 - 27 September 2007 XIV ECSMGE Madrid, Spain 21 - 24 October 2007 10 ANZ SMGE, Brisbane, Australia 7 - 9 November 2007 14th European Plaxis User Meeting Karlsruhe, Germany

Finite element method: the basics 5 - 10 May 2007 ITA-Aites World Tunnel Congress 2007 Prague, Czech Republic 8 - 11 May 2007 16th Southeast Asian Geotechnical Conference Selangor Darul Ehsan, Malaysia

26 - 28 November 2007 14 African Regional Conference SMGE, Yaounde, Cameroon-Africa

10 - 14 December 2007 13th Asian Regional Conference on Soil Mechanics and Geotechnical Engineering Calcutta, India

12 - 14 June 2007 Course Computational Geotechnics Manchester, United Kingdom

Prof. José E. Andrade Plaxis BV Department of Civil & Environmental Engineering

July, 2007

PO Box 572 2600 AN Delft The Netherlands Tel: +31 (0)15 251 77 20 Fax: +31 (0)15 257 31 07 E-Mail: [email protected] Website: www.plaxis.nl

7005910

25 - 28 June 2007 Course on advanced Computational Geotechnics Sydney, Australia

Outline • Applications of FEM • Fundamentals • Examples

Applications of FEM

patient-specific model

bypass alternatives

Biomechanics

!a

P

FOOTING

GRAIN SHEAR BAND

COMPACTIVE ZONE

!r

FAILURE SURFACE 'HOMOGENEOUS' SOIL

DILATIVE ZONE VOID

FIELD SCALE

LOG (m)

>1

SPECIMEN SCALE

0

-1

MESO SCALE

-2

Geomechanics

GRAIN SCALE

-3

DEVIATORIC STRAIN shear strain 0 3.

consolidation

0 .2 5

4

FE SOLN ANAL SOLN

0 2.

0 .1 5

0 1.

VERTICAL COORDINATE, m

3.5 3 2.5 2 1.5 1 0.5 0

0

20

40

60

80

PRESSURE, kPa

0 .0 5

Solid-fluid interactions

100

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 %!"!#$

)

&

)

'(1

'(1

'(0

'(0

'(/

'(/

'(.

'(.

'(-

'(-

'(,

'(,

'(+

'(+

'(*

'(*

'()

'& '

'()

'()

'(*

'(+

'(,

'(-

!!"!#$

'(.

'(/

'(0

'(1

)

'

FEM recipe Strong from Weak form Galerkin form Matrix form

Elastostatics: strong PDE B.C.’s

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

1

strong form

x

derived from continuum mechanics strong form = PDE + B.C.’s usually, there is no exact sln for strong form

Elastostatics: weak •Use principle of virtual work •Introduce virtual displacement w; •Use strong form !

w(1) = 0

1

w(u,xx +f ) dx = 0

0

!

0

1

w,x u,x dx =

!

0

1

wf dx + w(0)h

Elastostatics: Galerkin like g Construct approximate solution

uh = v h + g h like w

Construct functions based on shape functions wh =

n !

NA cA

vh =

A=1

n ! A=1

g h = gNn+1

NA dA

Piecewise linear FE 1 NA

N1

x1

x2

x A-1

xA

0

node

x A+1

Nn+1

xn

x n+1 1

finite element

x

Elastostatics: matrix Use weak form, plug-in Galerkin approximation n !

B=1

FA =

KAB dB = FA

!

KAB =

1

NA f dx + NA (0)h −

0

!

!

1

NA,x Nb,x dx

0

1

NA,x Nn+1,x dxg

0

it all boils down to...

K ·d=F stiffness matrix

force vector

Properties of K · d = F •Stiffness matrix is •symmetric •banded •positive-definite

may not always apply

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

Multi-D deformation ∇ · σ + f = 0 in Ω u = g on Γg σ · n = h on Γh

equilibrium e.g., clamp e.g., confinement

Γg

Constitutive relation given u get σ Ω

e.g., elasticity, plasticity Γh

FEM program TIME 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 triangular elements Gauss integration point displacement node

Modeling ingredients 1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C.’s 5. Solve

H4.417

B

Modeling ingredients 1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C.’s 5. Solve

Modeling ingredients 1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C.’s 5. Solve

Modeling ingredients !a

1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C.’s 5. Solve

!r

Modeling ingredients !a

1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C.’s 5. Solve

!r

Examples

Hyperbolic: LSST array

output input

Geometry and B.C.s output

input

Material parameters

shear modulus

degradation & damping

Input base acceleration

Acceleration output

Soil-fluid interaction Model ingredients

• • • •

Nonlinear continuum mechanics

!s " "#

Robust constitutive theory Computational inelasticity Nonlinear finite elements

x X

!f f

"#

x2 x1

Soil-fluid interaction Model ingredients

•

Nonlinear continuum mechanics

•

Robust constitutive theory

•

Computational inelasticity

•

Nonlinear finite elements

!’"=!’2=!’3

Soil-fluid interaction Model ingredients

•

Nonlinear continuum mechanics

•

Robust constitutive theory

•

Computational inelasticity

•

!$ Fn+1 Fn

!n

!"

Nonlinear finite elements

tr !n+1 !n+1

!#

Soil-fluid interaction Model ingredients

•

Nonlinear continuum mechanics

•

Robust constitutive theory

•

Computational inelasticity

•

Nonlinear finite elements

Displacement node Pressure node

Plane-strain compress specific volume

CT scan

FE model

Plane-strain compress specific volume

shear strain and flow

fluid pressure

Plane-strain compress

Questions?