finite element basics

7/29/2019 finite element basics

1/76

Introduction to the Finite Element Method

Sankara J. Subramanian

Department of Engineering DesignIndian Institute of Technology, Madras

December 11, 2009
http://goforward/http://find/http://goback/


2/76

Introduction tothe Finite

Element Method

Sankara J.Subramanian

Outline

What is FEM?

Basic

Formulation:Theory

Equilibrium

BoundaryConditions

ConstitutiveEquations

Kinematics

PVW

Basic

Formulation:Implementation

Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

1 What is FEM?

2 Basic Formulation: TheoryShape functionsFE matricesFE equations

3 Advanced FEMNonlinearityQuasi-continuum

4 Conclusion
http://find/http://goback/


3/76


Element Method


Outline

What is FEM?

Basic

Formulation:Theory

Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

The Finite Element Method is a numerical method

used for solving field problems

Origins in structural mechanics
http://find/


4/76


Element Method


Outline

What is FEM?

Basic

Formulation:Theory

Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion




Has strong mathematical foundation


5/76


Element Method


Outline

What is FEM?

Basic

Formulation:Theory

Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion




Has strong mathematical foundationWidely used by researchers structural and solid mechanics,fluid flow, heat transfer, electricity and magnetism and variouscoupled problems


6/76


Element Method


Outline

What is FEM?

BasicFormulation:Theory

Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion




Has strong mathematical foundationWidely used by researchers structural and solid mechanics,fluid flow, heat transfer, electricity and magnetism and variouscoupled problems

Routinely used in the industry design of buildings, airframes,

electric motors, automobiles, materials


7/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

Most problems of interest to engineers and

scientists cannot be solved analytically

From Projectile Impact on a Carbon Fiber Reinforced Plate, Abaqus

Technology Brief, Simulia Corporation, 2007
http://find/


8/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion





Plate is heterogeneous
http://find/


9/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion






Contact areas between steel ball and plate not known a priori
http://find/


10/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion







Failure criteria are complex
http://find/


11/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion







Failure criteria are complex

Analytical solution is out of the question!

G i diff i l i d
http://find/


12/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

Governing differential equations are converted to

algebraic equations

G i diff i l i d
http://find/


13/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion


algebraic equations

FEM is used to solve governing differential equationsapproximately

G i diff i l i d
http://find/


14/76


Element Method


Outline

What is FEM?

BasicFormulation:

Theory

Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion


algebraic equations


ODEs or PDEs are converted to a (large) system of algebraicequations, solved on computers

G i diff ti l ti t d t
http://find/


15/76


Element Method


Outline

What is FEM?

BasicFormulation:

Theory

Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion


algebraic equations



Unknown quantities are field variables (e.g. displacement,temperature) at nodes

G i diff ti l ti t d t


16/76


Element Method


Outline

What is FEM?

BasicFormulation:

Theory

Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion


algebraic equations




Quality of solution improves with increasing number of elements



17/76


Element Method


Outline

What is FEM?

BasicFormulation:

Theory

Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion


algebraic equations




Quality of solution improves with increasing number of elements

100 unknowns large in 1960s, now 106 unknowns routine!

We are interested in calculating the displacements
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18/76


Element Method


Outline

What is FEM?

BasicFormulation:

Theory

Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion


and stresses everywhere in a deforming elastic body

http://find/


19/76


Element Method


Outline

What is FEM?

BasicFormulation:

Theory

Equilibrium

BoundaryConditions


Kinematics

PVW

Basic


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



Elastic deformation is reversible = body returns to originalconfiguration when loads are removed

http://find/


20/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:Implementation

Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



Elastic deformation is reversible = body returns to originalconfiguration when loads are removedWe know the forces and displacements imposed on the body

http://find/


21/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



Elastic deformation is reversible = body returns to originalconfiguration when loads are removedWe know the forces and displacements imposed on the bodyForce F applied to a bar of cross-section A, stress = F/A

http://find/


22/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



Elastic deformation is reversible = body returns to originalconfiguration when loads are removedWe know the forces and displacements imposed on the bodyForce F applied to a bar of cross-section A, stress = F/A

For arbitrary 3-D solids, how do we compute thedisplacements and stresses everywhere in the body?

The elasticity boundary value problem is specified


23/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion


by the governing equation 1 , ...

The governing equation (no body forces, quasi-static case) is thelinear momentum balance equation:



24/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion




The stress tensor quantifies intensity of force (force/unit area) ata point.

http://find/


25/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion





11

x1 +

12

x2 +

13

x3 = 021x1

+ 22x2

+ 23x3

= 0 131x1

+ 32x2

+ 33x3

= 0

ij - stress tensor components,



26/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

e e ast c ty bou da y a ue p ob e s spec ed




11

x1 +

12

x2 +

13

x3 = 021x1

+ 22x2

+ 23x3

= 0 131x1

+ 32x2

+ 33x3

= 0


a system of 3 coupled PDEs for 6 unknown stress components

(ij = ji)

http://find/


27/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

y y p p




11

x1 +

12

x2 +

13

x3 = 021x1

+ 22x2

+ 23x3

= 0 131x1

+ 32x2

+ 33x3

= 0


a system of 3 coupled PDEs for 6 unknown stress components

(ij = ji)

ij,j = 0, i = 1, 3

b d di i 2
http://find/


28/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:

Implementation

Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

... boundary conditions 2 , ...

ijnj = Ti = Tspi on ST

ui = uspi on Su 2

ST and Su are the portions of external surface S on which forces anddisplacements are specified respectively.

i i i 3


29/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:

Implementation

Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

... constitutive equations 3 ...

Constitutive Equation - Hookes Law for isotropic, linear elasticity:

ij = E1 +

ij +

1 2kkij

3

E - Youngs modulus, - Poissons ratio, ij - strain tensorcomponents, ij - Kronecker delta

d ki ti l ti 4
http://find/


30/76


Element MethodSankara J.

Subramanian

Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:

Implementation

Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

... and kinematic relations 4 ...

Strain-displacement relations:

d ki ti l ti 4
http://find/


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Subramanian

Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:

Implementation

Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



ij =

1

2ui

xj +

uj

xi

4

and kinematic relations 4


32/76



Subramanian

Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:

Implementation

Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



ij

=1

2ui

xj+

uj

xi 4

11 =u1x1

reduces to the familiar definition of = l/l for the 1-D case.

The principle of virtual work is the basis of FEM
http://find/


33/76



Subramanian

Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:

Implementation

Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion


http://find/


34/76



Subramanian

Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:

ImplementationShape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion


The requirement of satisfying the governing PDE pointwise isrelaxed: we seek to satisfy it in a weak or integrated form.



35/76



Subramanian

Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



Instead of satisfying point-wise the equilibrium equationsij,j = 0, i = 1, 3 ,



36/76



Subramanian

Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



Instead of satisfying point-wise the equilibrium equationsij,j = 0, i = 1, 3 ,

we solve

V

ui ij,jdV = 0

Finite element equations follow from PVW
http://find/


37/76



Subramanian

Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion




38/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion


Using the divergence theorem and the strain-displacementrelations, we can restate the integral as

V

ijijdV =S

Fiu

j dS

where Fi are components of the applied loads on the boundary

http://find/


39/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

te e e e t equat o s o o o

Using the divergence theorem and the strain-displacementrelations, we can restate the integral as

V

ijijdV =S

Fiu

j dS

where Fi are components of the applied loads on the boundary

This is not an energy balance, it is merely a restatement ofequilibrium!

How is a finite-element solution implemented?


40/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

p

http://find/


41/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

p

First, the solid is discretized into a number of elements(meshing)

http://find/


42/76


Element Method


Outline

What is FEM?

BasicFormulation:

TheoryEquilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

p

First, the solid is discretized into a number of elements(meshing)

Nodes are typically points on the element boundaries

Displacement at any point in the solid is expressed

in terms of nodal displacements
http://find/


43/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion


The actual displacement components at any point inside anelement are interpolated from the (unknown) actual nodal valuesui


http://find/


44/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



The virtual displacements are also interpolated in the same way

u(x1, x2, x3) =

3i=1

Ni(x1, x2, x3)ui

Ni

are shape functions


http://find/


45/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



The virtual displacements are also interpolated in the same way

u(x1, x2, x3) =

3i=1

Ni(x1, x2, x3)ui

Ni

are shape functionsFor full 3D analysis, (3 # of nodes) unknowns

Linear shape functions are the simplest
http://find/


46/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

http://find/


47/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

u1(x1, x2) =4

i=1 Ni

(x1, x2)u2i1

u2(x1, x2) =4

i=1 Ni(x1, x2)u

2i

http://find/


48/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

u1(x1, x2) =4

i=1 Ni

(x1, x2)u2i1

u2(x1, x2) =4

i=1 Ni(x1, x2)u

2i

N1 = (ax1)(bx2)4ab N2 = (a+x1)(bx2)4ab

N3 = (a+x1)(b+x2)

4abN4 = (ax1)(b+x2)

4ab

Displacements are written in vector form


49/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

u1u2

=

N1 0 N2 0 N3 0 N4 00 N1 0 N2 0 N3 0 N4

u1

u2

u3u4

u5

u6

u7

u8

Derivatives of shape functions relate strains todisplacements
http://find/


50/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

displacements

Recall strain-displacement relations:

ij =1

2

uixj

+ujxi

In vector form,

=

112212

= [B]{ui}

where

[B] =

/x1 00 /x2

(1/2) /x1 (1/2) /x2

[N]

Matrix of elastic constants relates stresscomponents to strain components
http://find/


51/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

components to strain components

Recall Hookes law:

ij =E

1 + ij +

1 2

kkijIn matrix form, we can write this as

{} = [D]{} = [D][B]{ui}

where [D] is a matrix of elastic constants

Substitution into PVW gives element equilibriumequations
http://find/


52/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

equat o s

Recall the PVW:V

ijijdV =

S

Fiu

j dS

In matrix form,

V

{}T{}dV =

S

{u}T{F}dS

[K]{ui} = {R}

where

[K] =

V

[B]T[D][B]dV and {R} =

S

[N]T{F}dS

Assembly of element contributions yields globalstiffness equations
http://find/


53/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

q



54/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

q

Global equations are solved for the nodal displacements in the

entire solid

http://find/


55/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

q

Global equations are solved for the nodal displacements in the

entire solidUsing the finite-element matrices, strains and stresses are thencomputed

FEM is a versatile tool for studying nonlinear andcoupled phenomena
http://find/


56/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW

BasicFormulation:


FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

http://find/


57/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

Nonlinear material behaviour: plasticity, creep

http://find/


58/76


Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion


Large deformations: Geometric non-linearity

http://find/


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Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



Contact: bodies come into contact as a result of load application

http://find/


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Element Method


Outline

What is FEM?


Equilibrium

BoundaryConditions


Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion




Coupled phenomena: e.g. Piezoelectric ceramics, hydrogenembrittlement

http://find/


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Element Method


Outline

What is FEM?


EquilibriumBoundaryConditions


Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion




Coupled phenomena: e.g. Piezoelectric ceramics, hydrogenembrittlement

Fatigue and fracture studies

A finite-element formulation was used to studyhot-isostatic pressing of powders
http://find/


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Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

Bulk deformation: Elasticity + power-law creepInterparticle mass diffusionSurface diffusion

Modeling the interaction between densification mechanisms inpowder compaction (2001), S. J. Subramanian and P. Sofronis, Int.

J. Sol. Struct., 38, 7899-7918

FEM has recently been coupled with atomisticmodels
http://find/


63/76


Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

http://find/


64/76


Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

Stress-strain relations are derived from atomistic calculationsinstead of traditional continuum descriptions



65/76


Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion


Can capture the physics extremely accurately, but can only

model very small volumes (100s of nm3)



66/76


Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion




Mainly used in nanotechnology and materials researchapplications

http://find/


67/76


Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion





Field of active research

http://find/


68/76


Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion





Field of active research

http://www.qcmethod.com: a good resource

Quasi-continuum method used to study crackpropagation in Ni
http://find/


69/76


Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

Reference: Quasicontinuum simulation of fracture at the atomic scale(1998), R. Miller, E. B. Tadmor, R. Phillips and M. Ortiz, Modellingand Simulation in Materials Science and Engineering, vol. 6, 607-638

Conclusion and suggested references
http://find/


70/76


Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



71/76


Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion

FEM is an extremely valuable research tool that can be used ona wide variety of problems

http://find/


72/76


Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion


Hybrid FEM-atomistic computation is an active research areathat is growing rapidly

Textbooks:



73/76


Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



Textbooks:

Finite element procedures: K. J. Bathe



74/76


Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functions

FE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



Textbooks:


Concepts and applications of finite element analysis: R. D.Cook, D. S. Malkus, M. E. Plesha and R. J. Witt



75/76


Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functionsFE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



Textbooks:



The finite element method set: O. C. Zienkiewicz and R. L.

Taylor



76/76


Element Method


Outline

What is FEM?




Kinematics

PVW


Shape functionsFE matrices

FE equations

Advanced FEM

Nonlinearity

Quasi-continuum

Conclusion



Textbooks:



The finite element method set: O. C. Zienkiewicz and R. L.

Taylor

Thank you for your time

Documents

finite element basics