Previous class:

Introduction, course information, course outline, textbooks,

FEM: the big picture …

This class:

Introduction to FEM

“A Brief History of __________”

Advantages and Shortcomings of FEM

Misuses of FEA Programs: A Word of Caution

Textbook sections

Chapters 1 and 10

Chapter 10: Sections 1, 2, 8, 12, 13, 14, 15, 16, 17, 18

CEE570 / CSE 551 Class #2

1

FEM

PDE Song … (electronic)

Paper: ”P. S. Symonds and T. X. Yu, “Counterintuitive Behavior in a

problem of Elastic-Plastic Beam Dynamics” ASME Journal of Applied

Mechanics, Vol. 52, pp.517-522, 1985. (paper)

2 Handouts

2

Introduction to FEM

Civil Engineering

Aerospace Engineering

Materials Science

3 Etc, Etc, Etc …

Strong Wall in the Newmark Lab.

Finite element discretization (3D)

Strong Wall

Floor web

Slab 330 kips

990 kips330 kips

990 kips

330 kips

990 kips

Various critical loading cases

Real strong wall

(In courtesy of Prof. Kuchma and H. J. Lee)

Real strong wall

(Courtesy of Prof. Kuchma and H. J. Lee)

Results (vertical stress)

Civil Engineering Application

Finite element discretization and response of the pavement

Able to optimize the design before construction which can lead

to savings in the construction costs

Able to identify critical regions for the model

Able to observe detailed behavior of the model through the

various analysis, e.g. different loading conditions

(Courtesy of Dr. J.W. Kim)

5

Dynamic Fracture & Micro-Branching

Eran Sharon, Jay Fineberg, 1996. Microbranching instability and the dynamic fracture of brittle

materials, Physics Review B. Vol. 54, No. 10, pp. 7128-7193

Eran Sharon, Steven P. Gross, Jay Fineberg, 1996. Energy Dissipatiuon in Dynamic Fracture, Physics

Review Letters. Vol. 76, No. 12, pp. 2117-2120

Steel bars

Steel bars

PMMA plate

“seed” crack

Initial condition:

apply stress 10~18MPa

Fix upper and lower boundaries

I ntroduce sharp crack using a razor blade

Dynamic Fracture & Micro-Branching

Velocity

Fracture surface

microbranching

V<Vc V>=Vc V>Vc

Smooth single crack Branching pattern appears larger branches

Introduction to FEM

Method for numerical solution of field problems

Field Problem: Spatial distribution of dependent variables

Usually described by Differential Equation ( _______ form) or

Integral expression ( _____ form)

0, ,,,, gfueuducubuau yxyyxyxx

Finite element : Small piece of a structure

Standard PDE + Simple geometry/boundary condition

Analytical solution for the full field Calculus

Complicated PDE / Complex geometry /

boundary condition / material distribution:

Numerical solution for the full field FEM

9

strong

weak

Characterization of 2nd-order PDEs

2D 2nd order PDE: classified by the value of

• b2-4ac>0: __________

• b2-4ac=0: ________

• b2-4ac<0: ________

0, ,,,, gfueuducubuau yxyyxyxx

acb 42

(discriminant)

Classification not always simple

• Variable coefficients: equation type varies

• Coupled equation systems

• Higher dimensions

In General:

• Hyperbolic: time-dependent physical processes (e.g. wave motion) not

evolving towards a steady state

• Parabolic: time-dependent physical processes (e.g. heat diffusion) that are

evolving towards a steady state

• Elliptic: time-independent (steady state, equilibrium state) 10

hyperbolic

parabolic

elliptic

Finite element : Small piece of a structure

In the FEM, a complex region defining a continuum is discretized into simple geometric shapes called elements.

Continuum : infinite number of degrees-of-freedom (DOF),

Discretized model : finite number of DOF.

This is the origin of the name, finite element method.

Partial Differential Equations

System of Linear Equations (Ax=b)

Introduction to FEM

11

In each finite element field quantity :

A simple spatial variation u=f(x,y)

polynomials

Node, Element and mesh

Essence of FEM:

Approximation by piecewise

Interpolation of a field quantity

Introduction to FEM

Partial Differential Equations

System of Linear Equations (Ax=b)

12

Steps in modeling

FEA is simulation

Modeling error, Discretization error, Numerical error

13

History of FEM

14

It is difficult to document the exact origin of the FEM, because the

basic concepts have evolved over a period of 150 or more years.

The term finite element was first coined by Clough in 1960. In the

early 1960s, engineers used the method for approximate solution of

problems in stress analysis, fluid flow, heat transfer, and other

areas.

The first book on the FEM by Zienkiewicz and Chung was

published in 1967.

In the late 1960s and early 1970s, the FEM was applied to a wide

variety of engineering problems.

History of FEM

(Zienkiewicz)

15

History of FEM

Olgierd Zienkiewicz

John Argyris

Ray Clough

Bruce Irons

A Few FEM

Pioneers

The 1970s marked advances in mathematical treatments,

including the development of new elements, and convergence

studies.

Most commercial FEM software packages originated in the

1970s (ABAQUS, ADINA, ANSYS, MARK, PAFEC) and 1980s

(FENRIS, LARSTRAN ‘80, SESAM ‘80. )

History of FEM

(ANSYS multiphysics) 17

FEM Resources

Internet Finite Element Resources

Abaqus/Simulia

MSC/Patran

ANSYS

http://homepage.usask.ca/~ijm451/finite/fe_resources/fe_resources.html

http://staff.ttu.ee/~alahe/alem.html

www.abaqus.com

http://www.solid.ikp.liu.se/fe/index.html

www.mscsoftware.com/

www.ansys.com/

www.simulia.com

Advantages of FEM

• Can readily handle complex geometry:

• The heart and power of the FEM.

• Can handle complex analysis types:

• Vibration

• Transient analysis

• Nonlinear

• Heat transfer

• Fluids

• Can handle complex loading:

• Node-based loading (point loads).

• Element-based loading (pressure, thermal, inertial

forces).

• Time or frequency dependent loading.

• Can handle complex restraints:

• Indeterminate structures can be analyzed. 19

• Can handle bodies comprised of nonhomogeneous &

heterogeneous materials:

• Every element in the model could be assigned a different

set of material properties

• Functionally Graded Materials (FGMs)

• Can handle bodies comprised of anisotropic materials:

• Orthotropic

• Anisotropic

• Special material effects are handled:

• Temperature dependent properties.

• Plasticity

• Creep

• Swelling

• Special geometric effects can be modeled:

• Large displacements.

• Large rotations.

• Contact (gap) condition.

Advantages of FEM

(ANSYS multiphysics)

20

Shortcomings of FEM

A specific numerical result is obtained for a specific

problem. A general closed-form solution, which would

permit one to examine system response to changes in

various parameters, is not produced.

The FEM is applied to an approximation of the mathematical

model of a system (the source of so-called inherited errors.)

Experience and judgment are needed in order to construct a

good finite element model.

A powerful computer and reliable FEM software are

essential: the more elaborated the problem is, the more

computer power is needed.

21

• Numerical problems:

• Computers only carry a finite number of significant

digits.

• Round off and error accumulation.

• Susceptible to user-introduced modeling errors:

• Poor choice of element types.

• Distorted elements.

• Geometry not adequately modeled.

• Certain effects not automatically included:

• Buckling

• Large deflections and rotations.

• Material nonlinearities .

• Other nonlinearities.

Shortcomings of FEM

22

Misuse of FEA programs

Incorrect results caused damages

52 cases

Hardware error __

Software error __

User error __

Other causes __

“Computer Misuse – Are We dealing with a time bomb ? Who is to

blame and what are we doing about it ? A panel discussion,” in

Forensic Engineering, Proceedings of the First Congress, K. L. Rens

(Ed.), American Society of Civil Engineers, Reston, VA, 1997

23

7

13

30

2

Misuse of FEA programs

24

Curves 1-10: produced by FEM experts using

different commercial software packages.

Counterintuitive Behavior in a problem of

Elastic-Plastic Beam Dynamics

P.S. Symonds, T.X. Xu, 1985. Journal of Applied

Mechanics, ASME, Vol. 52, pp.517-522.

Beware of the pitfalls

of commercial software.

It is dangerous to use

FEM software blindly!!

Misuse of FEA programs

Message for users:

25

Next class

One Dimensional Elements (Ref.: CEE470)

One dimensional elements (bar, beam and frame elements)

General computational procedure

26