se207: validation of computational...

SE207 — Lecture V&V.04 — U N C L A S S I F I E D

U N C L A S S I F I E DXDIVISION

Engineering Institute Spring 2006 — LA-UR-05-6051

SE207: VALIDATION OF COMPUTATIONAL MODELS

LECTURE V&V.04

FINITE ELEMENT METHOD FUNDAMENTALS

François M. Hemez

Los Alamos National LaboratoryX-1.2, Mail Stop F699

Los Alamos, New Mexico 87545(E-mail: [email protected], Phone: 505-667-4631)

This presentation has been approved for unlimited, public release.Date: August 8, 2005Reference: LA-UR-05-6051Level: Unclassified

This material is to be used only for Los Alamos National Laboratory internal training or during the Los Alamos Dynamics Summer School. Distribution or reproduction requires prior approval. © Copyright 2003-2006 by François Hemez and the Los Alamos National Laboratory.




Pre-requisite• These notes are not meant to provide students with a

complete introduction to the finite element method.• Prior knowledge of the Finite Element Method (FEM) or

other numerical approaches in computational mechanics or computational physics is assumed.

• Students should also be somewhat familiar with basic techniques for solving Ordinary Differential Equations (ODE) and Partial Differential Equations (PDE).

• Students should be familiar with methods for numerical integration and finite differencing of equations.




Objectives• The first objective is to point to assumptions introduced

by computational approaches, such as the FEM, because understanding when these assumptions may break down is ultimately the focus of V&V activities.

• Because the main focus of this course is Structural Dynamics, the FEM is chosen. The same objective could be pursued with other computational techniques, such as finite volumes, finite difference schemes, etc.

• The second objective is to introduce a MATLABTM-based code for FE analysis, code that will be used throughout the course for applying various concepts of V&V.




Codes Provided• The first code is a MATLABTM-based finite element code.

The code is written to be a general-purpose, 3D code for linear elasticity. For simplicity, the version provided is restricted to linear spring and beam elements.

• The second code is a MATLABTM-based solver for finite differencing of the Burgers equation of non-linear wave propagation in 1D with smooth or shocked solutions.(#)

• The third code is a MATLABTM-based code for solving the 1D equations of hydro-dynamics in Lagrangian frame of reference. It implements compatible finite volumes to solve the conservation of mass, momentum, and energy without heat conduction. It is specialized to 1D geometry (Cartesian, cylindrical, spherical) and single-specie, ideal gas laws.(#)

(#) I am still working on this code and will most likely not make it available at this time …




Outline• Fundamentals of the finite element method

• A step-by-step example of the stiffness method

• Variational principles and minimization of total energy(#)

• A simple MATLABTM-based finite element code

(#) Not included in these notes. I will deal with this topic later when addressing discretization and the self-convergence of numerical solutions.




How Did This Get Started?• In the broad discipline of Solid Mechanics, large-scale

simulations are commonly performed with the FEM. How did this get started?

(Credit: ESA-WR at LANL.) (Credit: ESA-WR at LANL.)




Early History(1 of 2)

• What marks the “official” start of the FEM is the ~1955 development of “Structural Elements” at Boeing (Seattle, WA) to analyze the vibrations of delta wings. Two basic formulations are proposed, one based on the concept of stiffness, the other one based on a flexibility approach.

• Prior to the early 1960’s, these calculations were performed by hand using somewhat crude models, such as back-of-the-envelope calculation, 1D beam models, that nevertheless provided accurate-enough predictions.

• Why changing? New design concepts, such as delta or high aspect ratio wings, make it necessary to develop a general-purpose capability because stress distributions in structures can no longer be estimated with confidencefrom beam structure-like approximations.




Early History(2 of 2)

• The work of Professor Wilson at Berkeley (Department of Civil Engineering) make him what most consider today in the U.S. the “founding father” of the method. Professor Freijs de Veubeke at the University of Liège, Belgium, although not as well-known, is the other key figure.

• In 1963, engineers and applied mathematicians make the link between the Boeing “Structural Elements” and variational principles. The original FEM is re-discovered as a Ritz-Galerkin weak formulation of the equations of motion with piece-wise linear shape functions.

• In 1966, Carlos Felippa (now Professor at the University of Colorado at Boulder) authors his Ph.D. thesis on the calculation of stresses in dam structures using finite elements. It was, at the time, the 2nd or 3rd thesis ever published demonstrating what the method could do.




More Recent History• In the late 1960’s and throughout the 1970’s, the main

trend is for applied mathematicians to provide rigorous justification of the method and to prove its stability and convergence properties. The equivalence is established between strong and weak solutions.

• A few prominent names, among many others, are those of Professors Babushka, Belytschko, Brezzi, Hughes, Oden, Zienkiewicz, etc.

• More recently in the 1980’s, the French school of applied mathematicians (Ciarlet, Glovinski, Ladevèze, Lions, Nedelec, Ohayon, Raviart, etc.) build the theoretical foundations for many breakthroughs in disciplines such as a posteriori error indicators, fluid-structure interaction and acoustics, non-linear dynamics.




Basic Concept• The basic idea is that the equations of motion can only

be solved analytically, that is, solved exactly to provide a continuous solution, in the case of elemental geometries with simple-enough boundary conditions.

• One therefore decomposes a continuous problem into a multitude of similar, yet, discrete problems formulated on elemental geometries that are the finite elements.

Forces and Displacements Defined on a Continuous Domain

Forces and Displacements Defined on a Discrete Domain

Discrete Forces and Displacements Defined

on Elemental Geometries




Stiffness vs. Flexibility(1 of 2)

• The two formulations have yielded the stiffness method and the flexibility method. In the stiffness method, forces and moments are known and the equations of motion are solved to calculate the corresponding displacements and rotations.

• In the flexibility method, forces and moments are known and the (discrete) solution is obtained as a matrix-vector multiplication.

• Historically, both approaches were developed in parallel. McNeal, the Boeing engineer who was put in charge of the stiffness method, has moved on to eternal fame and glory. (You must have heard of MSC/NASTRAN, right?) Who remembers the name of the other guy tasked with developing the flexibility method?

[K] . u = f

u = [H] . f




Stiffness vs. Flexibility(2 of 2)

• The table below briefly summarizes the main attributes of the stiffness and flexibility methods. We will focus on the stiffness method in the remainder.

TrivialDifficultHandles rigid body modes?TrivialExpensiveCost of the numerical solveru = [H] . f[K] . u = fMain (linear) equation-of-motionChallengingTrivialAssembly procedureNot-so-easyRelatively EasyDerivation of element matricesFlexibility-likeStiffness-likeElement matrices

FlexibilityStiffnessCriterion

• It is interesting to note that, experimentally, what is measured during a vibration test is always related to a ratio between applied force and resulting displacement, which is, by definition, a flexibility …




Disassembly• Another technique is the somewhat obscure one know

as FE disassembly, as opposed to “assembly,” that links the stiffness and flexibility methods.

• The framework is similar to that of the stiffness method where the computational domain is decomposed into finite element and each element adds stiffness to the equations-of-motion.

• The main difference is that, instead of assembling the elements into a master stiffness, the elements are further disassembled into elemental strain-mode contributions. Strain-modes can be calculated analytically and they are used to calculate the flexibility matrix without requiring any full-size matrix inversion!

• Solving problems such as re-design, shape optimization, and FEM updating is very easy with a flexibility matrix.




An Example of Disassembly• Disassembly of an element stiffness matrix for the Euler-

Bernouilli 3D beam element in linear elasticity:

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

2EI00000

02

EI0000

002

EI000

0002

EI00

0000GJ000000EA

L1W

zz

zz

yy

yy

(e)

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−

−−

−−−

−−

=

00βγ00βγ0000

000010αα0000

00αα0000000100γβ00γβ0000000010αα000000αα00000001

Q(e)Local

( )T(e)Local

(e)(e)Local

(e)Local QWQk =

where α = (2√3)/L, β = (1 + √3), and γ = (1 – √3).

Reference: Hemez, F.M., Pagnacco, E., “Statics and Inverse Dynamics Solvers Based on Strain-Mode Disassembly,” European Journal of Finite Elements, Vol. 9, No. 5, June 2000, pp. 511-560. LA-UR-98-1359 and LA-UR-98-2502.




Steps of the FE Method• Step 1: Discretization of the computational domain

• Step 2: Idealization of the mechanics

• Step 3: Calculation of element matrices

• Step 4: Assembly of element matrices(#)

• Step 5: Application of boundary conditions

• Step 6: Resolution of the equations-of-motion

• Step 7: Calculation of secondary fields(#) Note that the logic of assembly (Step 4) may or may not apply depending on the type of solver implemented. Solvers such as explicit time integration (for non-linear dynamics) or conjugate gradient-based iterations (for linear problems or domain decomposition) can operate directly on the element-level matrices and they require no assembly.




Step 1: Discretization• Discretization addresses the choice of the N structural

elements and degrees-of-freedom that best characterize the problem. The basic principle of discretization is that the approximate solution of the discrete equations of motion converge to the exact solution of the continuousequations as N +∞.

Translations and Rotations:

Fluxes:

UX

UY

UZ

θX

θY

θZ

FN

FT1

FT2

• To converge one generally adds degrees-of-freedom, which increases the computational cost(#) (rule-of-thumb, solver cost grows as N2). Discretization also impacts the implementation of initial and boundary conditions.

(#) The Cholesky factorization K = L U costs N3/6 if K is symmetric and full; N.b2/2 if K is symmetric, sparse where “b” is the half bandwidth.




Step 2: Idealization• Idealization refers to the choice of mathematical models

to represent the mechanics or physics of interest.

Euler-Bernouilli Beam Theory

Un-deformed:

Neutral Axis

Neutral Axis

Deformed:

Timoshenko Beam Theory

Un-deformed:

Neutral Axis

Neutral Axis

Deformed:

• Implementing more realistic representations of reality is generally more difficult, costs more, but provides more “validity.” Idealization also impacts the implementation of initial and boundary conditions.




Step 3: Element Matrices• Element matrices embody the resolution of the discrete

equations of motion on an elemental (simple) geometry.

( ) (e)(e)Local

T(e)(e)Global

(e)By,

(e)Bx,

(e)y,A

(e)x,A

(e)Localk

(e)By,

(e)Bx,

(e)y,A

(e)x,A

(e)(e)Local

(e)

TkTk

UUUU

0000010100000101

LEA

FFFF

UkF

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

−+

⎟⎠⎞

⎜⎝⎛=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

444 3444 21Node A

Node B

X

Y

x,Ay,Aθ(e)

x,By,B

• Formulating the element matrices relies on assumptionsto describe the geometry, kinematics, material behavior, and the type of interpolation (known as shape functions).




Numerical Integration• Numerical integration is needed to integrate coefficients

of the master matrices over the volume of the element.

( ) ( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )∑ ∑ ∑

∫∫ ∫∫

= = =

+

−

+

−

+

−

≈

==

I J K

(e)

N1i N1j N1kkji

(e)(e)T(e)(e)Local

1

1

1

1

1

1

(e)(e)T(e)

Ω

(e)(e)(e)T(e)(e)Local

wwwγη;ξ;detJγη;ξ;Bγη;ξ;Cγη;ξ;Bk

γηξγη;ξ;detJγη;ξ;Bγη;ξ;Cγη;ξ;BdxBCBk

L L L

ddd

• Examples of 1D rules are given below. In general, a 1D integration rule based on NP points can integrate exactly a polynomial up to order (2 NP – 1).

( ) ( ) ( )

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+++⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

⎟⎠

⎞⎜⎝

⎛++⎟⎠

⎞⎜⎝

⎛−==

∫

∫∫+

−

+

−

+

−

106f

95f(0)

98

106f

95dξξf

31f

31fdξξf , 0f2dξξf

1

1

1

1

1

1




Step 4: Assembly• Assembly is the operation that “concatenates” values

contributed by each element into the master matrix.

A Bx,A

y,Ax,B

y,B

1 2

1

3

2

3

x,A

x,A

y,A x,B y,B

y,A

x,B

y,B

k(e) =Element

X1

X1

Y1 X2 Y2 X3 Y3

Y1

X2

Y2

X3

Y3

K =Master




Step 5: Boundary Conditions• Boundary conditions specify how the structure should

behave at the “edge” of the computational domain. The main two types of conditions for FE, structural analysis are the Dirichlet and Neumann boundary conditions.

• Applying the boundary condition can be as simple as “deleting” equations that correspond to the prescribed displacements from the master matrices.(#) It may also be excruciatingly difficult when dealing with problems that involve coupled physics.

Applied Forces

Prescribed Displacements

(#) “Deleting” is shown between quotes because equations that correspond to a prescribed displacement are never actually deleted. They are simply flagged and by-passed when assembling the master matrices.




Step 6: Resolution• After the equations of motion have been discretized, they

are solved to provide a solution for the primary field variables, generally, written as displacement and rotation degrees-of-freedom.

• The master equations of motion for linear statics, linear dynamics, and generic, non-linear dynamics are shown:

• Solving the discretized equations of motion generally involves a direct solver that factors a master matrix; an iterative solver such as conjugate gradient; a non-linear solver such as Newton-Raphson; or time integration.

( ) ( )

( ) ( ) (t)FttUU(t);F

ttUM:linearNon

(t)FKU(t)ttUD

ttUM , FKU:Linear

ExtInt2

2

Ext2

2

Ext

=⎟⎠⎞

⎜⎝⎛

∂∂+

∂∂−

=+∂

∂+∂

∂=




Time Integration• Time integration schemes are implemented to advance

the solution fields in time when solving time-dependent (dynamic) systems of equations.

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( )

( ) ( )tx;UΔt21tx;U2Δttx;U with

Δt

Δt21t-x;UΔt

21tx;U

ttx;U

:Rule lTrapezoida

Δt2Δtt-x;Utx;U2Δttx;U

ttx;U:Difference Central

ΔtΔtt-x;Utx;U

ttx;U:Backward Euler

Δttx;UΔttx;U

ttx;U:Forward Euler

−⎟⎠⎞

⎜⎝⎛ +=+

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛ +

≈∂

∂

+−+≈∂

∂

−≈∂

∂

−+≈∂

∂

• Verifying the quality of the (discrete in time) solution can be extremely difficult and it is often an act of delicate balancing between stability, accuracy, and the available computational resource.




Step 7: Secondary Fields• After the primary fields are solved for, the solutions can

be converted back into element-level quantities, from which secondary fields, such as stresses, are calculated.

( )(e)Local

(e)(e)Local

(e)Local

(e)(e)Local

(e)Global

(e)(e)Local

εCσ:Equation veConstituti levelElement-UBε:Solution Strain levelElement-UTU:ntsDisplaceme localto-Global-

==

=

• In the case, for example, of a linear, elastic bar element, the local displacement vector is formed from those at the two extremities U(e) = (UA; UB)T; the displacement-to-strain differentiation operator reduces to the row-vector B(e) = (1/L; –1/L) where L is the length of the bar; and the constitutive model of linear elasticity is σ(e) = E . ε(e).

“The bottom line is that one makes many assumptions here as well!”




Where Are V&V Activities Important?• Step 1: Discretization ……………… Solution convergence

• Step 2: Idealization …………… Validation of assumptions

• Step 3: Element matrices ………….……. Code verification

• Step 4: Assembly …………………….… Nothing significant

• Step 5: Boundary conditions .…..…. Nothing significant(#)

• Step 6: Resolution …………………. Solution convergence

• Step 7: Secondary fields .……. Validation of assumptions

(#) V&V activities are not particularly relevant to boundary conditions unless sub-models are involved to define them. Examples are models to describe the behavior of acoustic fields, wave propagation, or radiated energy far away from the domain of interest.




A Simple Example• The main objective of this example is to show how to

write the equations that relate global displacements to global forces for a simple frame structure.

• The example illustrates the two foundational principles of: 1) compatibility of displacements; and 2) balancing of internal and external forces. (Note that “compatibility” of displacements does not mean “continuity.”)

• For historical and pedagogical reasons, this presentation follows derivations of the conventional stiffness method.




Definition of the Problem• The example is the analysis of a simple frame structure

with one extremity of the horizontal load-bearing member grounded and the other one supported by a sliding condition.

UX

UY

FX

FY

X

Y

Grounded Sliding

Node 3

Node 2Node 1

• The goal of what follows is to derive equations that relate the global displacements and applied forces in statics or dynamics.




Discretization• The frame is discretized with three structural elements.

+135 degreesNode 3Node 1FE 3+90 degreesNode 3Node 2FE 2+0 degreesNode 2Node 1FE 1AngleNode BNode AElement

1 2

1

3

2

3

Element 1

Element 2Element 3

X

Y

Grounded Sliding

Node 3

Node 2Node 1

X

Y

Angle θ(Defined From

Node A to Node B)

Node B

Node A




Idealization• The mechanics is idealized using bar elements with axial

deformation only; this is an assumption. The elasticity is assumed to be linear, which means that the stress-strain constitutive relation reduces to a single constant, E, that is the modulus of elasticity; this is another assumption.

( ) ( )

( ) ( )( ) ( ) ( )

( )BA

BA

BA

UUL

EAConstantF

xσxAxFxEεxσ

elongation an for 0ε that convention the withL

UUConstantε

LUU

dxxdUxε

−⎟⎠⎞

⎜⎝⎛==

==

<

−=≈

−≈=

x(e)

Node BNode AElement “e”

y(e)

UBFB

UA

FA

• A local frame of reference is defined for each element, with a local axis x(e) oriented from Node A to Node B. The forces, displacements, strains, and stresses given in the local frame of reference.




Degrees-of-freedom• The discretization and idealization result in a total of six

degrees-of-freedom. There is no rotation or other type of degree-of-freedom because of the idealization chosen.

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=

==

=−−−−

Y,3

X,3

Y,2

X,2

Y,1

X,1

1by6Global

Y,3

X,3

Y,2

X,2

Y,1

X,1

1by6Global

FFFFFF

F ,

UU

0UU

0U0U

U

1(UX,1;FX,1)

Element 1

Element 2Element 3

X

Y

(UX,2;FX,2)

(UX,3;FX,3)

(UY,1;FY,1) (UY,2;FY,2)

(UY,3;FY,3)

• The forces are reaction forces at the prescribed degrees-of-freedom, FX,1, FY,1, and FY,2. The forces are known and equal to the external forces at the “free” degrees-of-freedom, FX,2, FX,3, and FY,3.




The Element Freedom Table• The Element Freedom Table (EFT) is a table that keeps

tract of which (global) degrees-of-freedom is contributed by each finite element. Doing so established a mapping between the local and global degrees-of-freedom.

1

2

3

4

X

Y

Grounded0 0 +10 +1+1

Node A Node B Node C Node D

Element

1

+1 -1 +1+1 -1 +1 +1+12

+1 +1 +1+1 +1 +1+13

+1 -1 +1+1 -1 +1 +1+14

-1

-1

+1

+1

UY θZUX UY θZUX UY θZUX UY θZUX

+1 -1

+1 -1

+1+1

Legend: “-1” = Inactive; “0” = Prescribed; “+1” = Active.




Local-to-global Frames(1 of 2)

• A rotation operator must be defined to convert forces and displacements expressed in the local element frame of reference (x(e);y(e)) into quantities defined in the global coordinate system (X;Y).

( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( )⎥

⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=

==−

(e)(e)

(e)(e)

(e)(e)

(e)(e)

(e)

(e)Local

T(e)(e)Global

(e)Local

1(e)(e)Global

θcosθsin00θsinθcos0000θcosθsin00θsinθcos

T

FTF , UTU

x(e)

y(e)

UY,B UB

UX,B

UA

X

Y

Angle θ(e)

(Defined From Node A to Node B)

Node B

Node A

UY,A

UX,A




Local-to-global Frames(2 of 2)

(e)Global

(e)Global

(e)Global

(e)BY,

(e)BX,

(e)Y,A

(e)X,A

(e)Global

(e)BY,

(e)BX,

(e)Y,A

(e)X,A

(e)Global

(e)Local

(e)Local

(e)Local

(e)BY,

B(e)

BX,

(e)Y,A

A(e)X,A

(e)Local

(e)By,

B(e)

Bx,

(e)y,A

A(e)x,A

(e)Local

UkF

FFFF

F ,

UUUU

U

UkF

0FFF0FFF

F ,

0UUU0U

UU

U

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

====

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

===

=

=

( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )⎥

⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=

==

=−

(e)(e)

(e)(e)

(e)(e)

(e)(e)

(e)

(e)Local

T(e)(e)Global

(e)Local

1(e)(e)Global

(e)(e)Local

T(e)(e)Global

θcosθsin00θsinθcos0000θcosθsin00θsinθcos

T

FTF , UTU

TkTk

x(e)

y(e)

UY,B UB

UX,B

UA

X

Y

Angle θ(e)

(Defined From Node A to Node B)

Node B

Node A

UY,A

UX,A




The Structural Element(1 of 2)

• Now that the frame is decomposed in elements, and that local frames are defined for each element, the equations of motion can be written within each element.

( ) ( )

( )

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+−−+

⎟⎠⎞

⎜⎝⎛=

⎭⎬⎫

⎩⎨⎧

==

−⎟⎠⎞

⎜⎝⎛==

−≈=

B

A

B

A

BA

BA

BA

UU

1111

LEA

FF

FFF

UUL

EAConstantF

LUU

dxxdUxε x(e)


y(e)

UBFB

UA

FA

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

===

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

−+

⎟⎠⎞

⎜⎝⎛=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

====

0UUU0U

UU

0000010100000101

LEA

0FFF0FFF

(e)By,

B(e)

Bx,

(e)y,A

A(e)x,A

(e)BY,

B(e)

BX,

(e)Y,A

A(e)X,A




The Structural Element(2 of 2)

• The axial displacement-only (bar) element is analogous to a spring element.

( ) ( )

( )

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+−−+

⎟⎠⎞

⎜⎝⎛=

⎭⎬⎫

⎩⎨⎧

==

−⎟⎠⎞

⎜⎝⎛==

−≈=

B

A

B

A

BA

BA

BA

UU

1111

LEA

FF

FFF

UUL

EAConstantF

LUU

dxxdUxε

x(e)


y(e)

UBFB

UA

FANode A

UBUA

Node B

+f –f

u

( )

( )AB

AB

UULkkεF

kεConstantF :Force InternalUUConstantδU :Elongation

LδUConstantε :Strain Small""

−⎟⎠⎞

⎜⎝⎛==⇒

≈=−==

≈=




Rotation of Elements• Equations of the previous 4 view-graphs are combined to

rotate the local stiffness matrix from the (local) frame of reference (x(e);y(e)) to the coordinate system (X;Y).

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

−+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−⎟⎠⎞

⎜⎝⎛==

CS00SC0000CS00SC

0000010100000101

CS00SC0000CS00SC

LEATkTk

T

(e)(e)Local

T(e)(e)Global

where C = cos(θ(e)) and S = sin(θ(e)).

• This step is pure “machinery” and it is the same for other types of finite elements, whether in 1D, 2D, or 3D.




Assembly(1 of 5)

• Now that each element can be written in the global coordinate system, it is ready for assembly. “Assembly” means that two fundamental rules are combined: the compatibility of displacements and balancing of forces.

Rule 1: UX,A(1) = UX,A

(2) = UX,A(3) = UX,AElement 1 Element 2

Element 3Node A Rule 2: FX,A

(1) + FX,A(2) + FX,A

(3) = FX,A(Ext)

• Displacements contributed by different elements at a same nodal joint / degree-of-freedom are equal: Rule 1.

• The summation of forces contributed by elements that all share a common nodal joint / degree-of-freedom are equal to the external (applied) forces: Rule 2.




Assembly(2 of 5)

• Application of the two rules to the frame discretization:

⎩⎨⎧

====

⎩⎨⎧

===

⎩⎨⎧

==

Y,3(3)Y,3

(2)Y,3

X,3(3)X,3

(2)X,3

Y,2

X,2(2)X,2

(1)X,2

Y,1

Y,1

UUU: YDirection in 3 Node at 1 RuleUUU: XDirection in 3 Node at 1 Rule

0U: YDirection in 2 Node at 1 RuleUUU: XDirection in 2 Node at 1 Rule

0U: YDirection in 1 Node at 1 Rule0U: XDirection in 1 Node at 1 Rule

⎩⎨⎧

=+=+

⎩⎨⎧

=+=+

⎩⎨⎧

=+=+

(Ext)Y,3

(3)Y,3

(2)Y,3

(Ext)X,3

(3)X,3

(2)X,3

(Reaction)Y,2

(2)Y,2

(1)Y,2

(Ext)X,2

(2)X,2

(1)X,2

(Reaction)Y,1

(3)Y,1

(1)Y,1

(Reaction)X,1

(3)X,1

(1)X,1

FFF: YDirection in 3 Node at 3 RuleFFF: XDirection in 3 Node at 3 Rule

FFF: YDirection in 2 Node at 2 RuleFFF: XDirection in 2 Node at 2 Rule

FFF: YDirection in 1 Node at 2 RuleFFF: XDirection in 1 Node at 2 Rule 1 2

1

3

2

3

Element 1

Element 2Element 3




Assembly(3 of 5)

• The application of the two rules leads to the assembly of the element-level equations into master equations.

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )GlobalGlobal

31e

(e)Global

GlobalGlobal(3)GlobalGlobal

(2)GlobalGlobal

(1)Global

GlobalGlobal(3)GlobalGlobal

(2)GlobalGlobal

(1)Global

Master(3)Global

Master(2)Global

Master(1)Global

(Ext)Global

(3)Global

(3)Global

(2)Global

(2)Global

(1)Global

(1)Global

(Ext)Global

(3)Global

(2)Global

(1)Global

Global(3)Global

(2)Global

(1)Global

(3)Global

(3)Global

(3)Global

(2)Global

(2)Global

(2)Global

(1)Global

(1)Global

(1)Global

FUk

FUkUkUk

FUkUkUk1 Rule

UUUUUU

FUkUkUkFFFF

FFFF2 Rule

UkFUkFUkF

=⎟⎠

⎞⎜⎝

⎛

=++

=++⎪⎭

⎪⎬

⎫

=

=

=

=++

=++

=++⎪⎭

⎪⎬

⎫

=

=

=

∑

⇒

⇒

= L




Assembly(4 of 5)

• The assembly procedure is pictured graphically below.

1 2

1

3

2

3

x,A

x,A

y,A x,B y,B

y,A

x,B

y,B

k(e) =Element

X1

X1

Y1 X2 Y2 X3 Y3

Y1

X2

Y2

X3

Y3

K =Master




Assembly(5 of 5)

• The equations for the general case of 3D, linear, static elasticity are summarized below in which Ne denotes the total number of elements in the computational mesh.

( ) ( )

( )

( ) ( ) (e)

Ω

(e)(e)(e)T(e)

N1e

T(e)Master

Ω

(e)(e)(e)T(e)(e)Local

Ω

(e)(e)Local

T(e)Local

(e)Local

(e)Local

T(e)Local

(e)Local

(e)(e)Local

(e)Local

(e)(e)Local

TdxBCBTK

dxBCBk

dxσεUkU

εCσUBε

(e)e

(e)

(e)

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

=

=

=

∫∑

∫

∫

= L

( ) (e)

N1e

(e)Local

T(e)

N1e

(e)GlobalMaster

GlobalGlobalMaster

TkTkK

FUK

ee

∑∑==

==

=

LL




Assembly in MATLABTM

• In MATLABTM, the assembly of element-level equations into master matrices can be implemented in two ways.

• The first strategy is to add the element stiffness matrix to a subset of the master stiffness, using its partition that corresponds to the same degrees-of-freedom.

• The second strategy is to expand the element stiffness matrix into a storage of same size as the master stiffness matrix, then add the two matrices together.

• The first option is more intuitive. The second option is computationally much more efficient when dealing with large FE models (lots of finite elements), provided that all arrays are stored sparse.




Boundary Conditions• Applying boundary conditions such as prescribed

displacements or rotations simply consists of deleting the corresponding rows and columns in the equations.

X1

X1

Y1 X2 Y2 X3 Y3

Y1

X2

Y2

X3

Y3

KMaster

X1

Y1

X2

Y2

X3

Y3

=

X1

Y1

X2

Y2

X3

Y3

UGlobal FGlobal =• Boundary conditions for the frame discretization are displacements

UX = UY = 0 at Node 1, and UY = 0 at Node 2.




Resolution• The system of master equations is solved, and boundary

conditions are used to calculate the reaction forces.

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

••••••••••••••••••

=

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

Y,3

X,3

X,2

666563

565553

464543

363533

262523

161513

(Ext)Y,3

(Ext)X,3

(Reaction)Y,2

(Ext)X,2

(Reaction)Y,1

(Reaction)X,1

UU

0U

00

KKKKKKKKKKKKKKKKKK

FF

FF

FF

• Here, one does not care about the columns 1, 2 and 4 of the master stiffness matrix because these stiffness values multiply zero-valued displacements. If a prescribed displacement is non-zero, then it will contribute a non-zero force and, of course, the corresponding column of the stiffness matrix must be taken into account.

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧−

(Ext)Y,3

(Ext)X,3

(Ext)X,2

1

666563

565553

363533

Y,3

X,3

X,2

FFF

KKKKKKKKK

UUU




Code Provided• The code provided is a simplified version of a general-

purpose package for Research and Development in finite element methodologies, written by F. Hemez.

• Handles arbitrary 3D geometry, but restricted to linear problems (linear material, small strain and displacement, no contact) and a few elements (bar, beam, 3-node plate, 8-node brick, translational and rotational constraints).

• Solvers are implemented for static or frequency-domain analyses, time-domain integration, finite element model updating, and finite element disassembly.

• The code provided is greatly simplified by removing the sparse storage functionality and specializing the element library to 2-node spring and beam elements.




Architecture• The architecture of the code follows the following steps:

%...Define the problem (coordinates, topology, materials, etc.)FEMDefinition = struct(‘coordinates’,Coordinates,‘topology’,Topology, …);%...Assembly of the master mass and stiffness matrices[M,K] = assembly(FEMDefinition);%...Extract the eigen-vectors and corresponding eigen-values[EigenVectors,EigenValues] = eig(K,M);%...Convert the eigen-values into frequencies (in units of Hertz)Frequencies = sqrt(diag(EigenValues))/2/pi;%...Print the frequencies on screendisp(Frequencies);

• To define a problem, one must enter the information in a format specified by the structure array FEMDefinition.

• To implement a new element type, one must modify the m-function assembly.m (and nothing else).




The LANL 8-DOF System• The code is used to illustrate the assembly of master

mass and stiffness matrices for the LANL 8-DOF system.

• The system is an assembly of eight masses connected by seven springs and free to slide on a central rod that provides support for the assembly.




Idealization• The LANL 8-DOF system is idealized as a series of eight

masses connected by seven linear springs. Boundary conditions are assumed to be free-free.

• These choices (idealization and discretization) result in a total of eight translational degrees-of-freedom X1 … X8.

M3

k2

U3

M4

k3

U4

M5

k4

U5

M6

k5

U6

M2

k1

U2

M7

k6

U7

M8

k7

U8

M1

U1

(Shaker attached at mass M1.) (Positive displacements in this direction.)X-axis




Assumptions Made• Three-dimensional geometry simplified as 1D.• Behavior of springs assumed linear.• Spring stiffness values assumed known.• Friction of masses on the central support rod neglected.• Potential preloading neglected.• Effect of gravity neglected.• Boundary condition idealized as free-free.• Feed-back between shaker and system neglected.• … And probably a few others that do not, however, have

much influence on the prediction of modal frequencies and mode shape vectors.




Definition of a Model• A model is defined by populating a structure array called

FEMDefinition. Users must comply with the format defined by FEMDefinition and its contributing variables.

%...Definition of a structure array to store the FEM definitionFEMDefinition = [ ];FEMDefinition = setfield(FEMDefinition,‘coordinates’,Coordinates);FEMDefinition = setfield(FEMDefinition,‘topology’,Topology);FEMDefinition = setfield(FEMDefinition,‘attributes’,Attributes);FEMDefinition = setfield(FEMDefinition,‘materials’,Materials);FEMDefinition = setfield(FEMDefinition,‘masses’,LumpedMasses);FEMDefinition = setfield(FEMDefinition,‘frames’,ElementFrames);FEMDefinition = setfield(FEMDefinition,‘bconditions’,BoundaryConditions);

%...Assembly of the master mass and stiffness finite element matrices[M,K] = assembly(FEMDefinition);




Dimensions• To get started, look at the main driver m-file Spring8.m. It

starts defining the problem by specifying dimensions such as the number of nodes, number of elements, etc.

%...Define the number of nodesnNodes = 8;%...Define the number of finite elements (springs)nElements = 7;%...Define the number of materialsnMaterials = 2;%...Define the number of lumped (discrete) massesnLMasses = 8;%...Define the number of displacement boundary conditionsnBConstraints = 16;

• These are used to define the dimensions of arrays that store the nodal coordinates, define the elements, etc.




Coordinates• Coordinates are defined as a triplet (X;Y;Z) for each node

of finite element discretization. The number of rows of array Coordinates is equal to the number of nodes.

Coordinates = zeros(nNodes,3);Coordinates = [ 1.0, 0.0, 0.0; ...

2.0, 0.0, 0.0; ...3.0, 0.0, 0.0; ...4.0, 0.0, 0.0; ...5.0, 0.0, 0.0; ...6.0, 0.0, 0.0; ...7.0, 0.0, 0.0; ...8.0, 0.0, 0.0];

• Be careful to enter numerical values (starting with the nodal coordinates) expressed in units that are consistent with the units of other physical quantities defined.

Note that, for something this simple, the nodal coordinates can be defined using a compact statement such as:

Coordinates = zeros(nNodes,3);Coordinates(:,1) = [1:nNodes]’;




Topology• The topology array Topology defines the list of nodes for

each finite element of the discretization. Its number of rows is equal to the number of elements.

Topology = zeros(nElements,2);Topology = [ 1, 2; ...

2, 3; ...3, 4; ...4, 5; ...5, 6; ...6, 7; ...7, 8];

• Node identifiers are equal to the row of array Coordinatesat which the node is defined. (For example the 5th node is defined on the 5th row of array Coordinates.)




Attributes• The attribute array Attributes is a two-column matrix that

stores two identifiers for each finite element.

• The first identifier (first column) is a flag that defines the element type. The second identifier (second column) is a flag that defines the material and geometrical properties.

%...Damage case 5 where the stiffness of spring #5 is reduced by 14%Attributes = [ 1, 1; ...

1, 1; ...1, 1; ...1, 1; ...1, 2; ...1, 1; ...1, 1];

Definition of element types:

Type-1, Two-node spring (bar)Type-6, Two-node beamType-8, Three-node plateType-17, Eight-node brick




Material Groups(1 of 2)

• The material array Materials is an eight-column matrix that stores the material and geometrical properties of elements. Each row corresponds to one of the material groups defined in array Attributes.

• Two material types are defined above. Material type “1” (on the first row) is for undamaged springs and material type “2” (on the second row) is for damaged springs.

%...Initialization of material propertiesMaterials = zeros(nMaterials,8);Materials = [ Area_U, E_U, Nu_U, Rho_U, H_U, Ixx_U, Iyy_U, Izz_U; ...

Area_D, E_D, Nu_D, Rho_D, H_D, Ixx_D, Iyy_D, Izz_D];

Column 1: Cross-sectional area, Area; Column 2: Modulus of elasticity, E; Column 3: Poisson’s ratio, Nu; Column 4: Density, Rho; Column 5: Plate or shell thickness, H; Columns 6-to-8: Beam moments of inertia, (Ixx;Iyy;Izz).




Material Groups(2 of 2)

• Spring stiffness coefficients are calculated using unit cross-sectional areas A = 1, unit coordinates L = 1, and a modulus of elasticity equal to the spring stiffness k such that the equation k = E.A/L is verified.

%...Material properties for an undamaged springArea_U = 1;E_U = 322.0e+06;Nu_U = 0;Rho_U = 0;H_U = 0;Ixx_U = 0;Iyy_U = 0;Izz_U = 0;

%...Material properties for a damaged springArea_D = Area_U;E_D = 0.86*E_U;Nu_D = Nu_U;Rho_D = Rho_U;H_D = H_U;Ixx_D = Ixx_U;Iyy_D = Iyy_U;Izz_D = Izz_U;

Irrelevant material or geometrical properties are set to zero (including, here, the density because masses are defined as point masses).

Definition of a 14% stiffness reduction




Lumped Masses• Lumped masses are defined in the two-column matrix

LumpedMasses. They are point-mass values added to all translational degrees-of-freedom of the master mass matrix at nodes specified in the first column.

LumpedMasses = zeros(nLMasses,2);LumpedMasses = [ 1, 3231.1; ...

2, 2423.0; ...3, 2421.3; ...4, 2423.0; ...5, 2422.5; ...6, 2422.5; ...7, 2421.3; ...8, 2423.0];

Nodes at which a point-masses are added in all translational directions (UX;UY;UZ).

Values of the point-masses.




Local Frames• Local element frames-of-reference defined using the

nine-column array ElementFrames are used to rotate beam elements in the global coordinate system (X;Y;Z).

ElementFrames = zeros(nElements,9);ElementFrames = [ 1, 0, 0, 0, 1, 0, 0, 0, 1; ...

1, 0, 0, 0, 1, 0, 0, 0, 1; ...1, 0, 0, 0, 1, 0, 0, 0, 1; ...1, 0, 0, 0, 1, 0, 0, 0, 1; ...1, 0, 0, 0, 1, 0, 0, 0, 1; ...1, 0, 0, 0, 1, 0, 0, 0, 1; ...1, 0, 0, 0, 1, 0, 0, 0, 1];

X

Y

Z

UV

WNode A

Node B

Local frames are used with beam elements only. Four important notes: 1) The triad (U;V;W) must be orthonormal; 2) The first local frame U always points from Node A (first node defined in Topology) to Node B (second node defined in Topology); 3) The second local frame V is associated to the moment of inertia Iyy; 4) The third local frame W is associated to the moment of inertia Izz.




Prescribed Displacements• The boundary condition is defined in terms of prescribed

displacements at specific nodes and specific directions in the three-column array BoundaryConditions. (No other type of boundary condition is currently allowed.)

BoundaryConditions = zeros(nBConstraints,3);BoundaryConditions = [ 1, 2, 0; ...

1, 3, 0; ...2, 2, 0; ...2, 3, 0; ...

etc. etc. etc.

8, 2, 0; ...8, 3, 0];

Degrees of freedom are numbered:

1, Translation UX in the X-direction2, Translation UY in the Y-direction3, Translation UZ in the Z-direction4, Rotation θX around the X-axis5, Rotation θY around the Y-axis6, Rotation θZ around the Z-axis

Column 1 gives the node number

Column 2 gives the degree-of-freedom number

Column 3 gives the value of the prescribed displacement or rotation




Assembly(1 of 3)

• Once the finite element model is defined with the structure array FEMDefinition, the master matrices are assembled using the m-file assembly.m.

%...Assembly of the master mass and stiffness finite element matrices[M,K] = assembly(FEMDefinition);

• Assembly of the finite elements follows five basic steps:Step 1: Extract the data from structure array FEMDefinitionStep 2: Calculate the Element Freedom Table (EFT)Step 3: Loop on finite elements,

Step 3a:Form the element mass matrix MeStep 3b:Form the element stiffness matrix KeStep 3c:Merge M M + Me and K K + Ke

End loop on finite elementsStep 4: Add the lumped massesStep 5: Apply the boundary condition (irrelevant here)

for e

=1…

NE




Assembly(2 of 3)

• The boundary condition needs not be applied (that is, the master matrices are not modified by deleting rows and columns) because prescribed degrees-of-freedom are flagged by the Element Freedom Table (EFT) and these equations are automatically avoided during assembly.

%...Compact the element freedom table and element matricesListActiveDOF = find(ElementEFT>0);ElementEFT = ElementEFT(ListActiveDOF);Me = Me(ListActiveDOF,ListActiveDOF);Ke = Ke(ListActiveDOF,ListActiveDOF);

Line 1: Find the indices of active degrees-of-freedom (that are strictly positive).Line 2: Restrict the EFT to active degrees-of-freedom only.Line 3: Restrict the element mass Me to active equations only.Line 4: Restrict the element stiffness Ke to active equations only.Only equations corresponding to active degrees-of-freedom are assembled in the master matrices, and there is no need to apply the boundary condition afterwards.




Assembly(3 of 3)

• The first option to merge an element matrix into a master matrix is to operate on the subset of rows and columns of the master matrix that correspond to the element.

K(ElementEFT,ElementEFT) = K(ElementEFT,ElementEFT) + Ke;

• The second option to merge an element matrix into a master matrix is to expand the element matrix to the full size of the master matrix.

• Option 2 is computationally more efficient when dealing with large numbers of finite elements (NE ≥ 10,000).

nFEentries = length(ElementEFT)^2;RowIndices = diag(ElementEFT)*ones(size(Ke));ColIndices = (diag(ElementEFT)*ones(size(Ke)))';RowIndices = (RowIndices(:))’;ColIndices = (ColIndices(:))’;K = K + sparse(RowIndices,ColIndices,Ke(:),nDOF,nDOF,nFEentries);




Master Matrices (M;K)• Run the driver Spring8.m and verify that the master

stiffness matrix for the undamaged model given by:

-322-322000000-322644-32200000

0-322644-322000000-322644-322000000-322644-322000000-322644-322000000-322644-322000000-322322

• Verify that the master mass matrix for the undamaged model is a diagonal matrix whose main diagonal is:

KMaster = 10+6 .

2423.02421.32422.52422.52423.02421.32423.03231.1MMaster =




Resonant Frequencies• Likewise, verify that the resonant frequencies of the

undamaged mass-spring model are equal to the calculated values listed below.

+13.79%113.71 Hertz131.90 Hertz7-0.45%113.71 Hertz113.20 Hertz6+4.37%95.63 Hertz100.00 Hertz5+5.96%80.78 Hertz85.90 Hertz4+2.83%62.97 Hertz64.80 Hertz3+2.01%43.02 Hertz43.90 Hertz2+2.29%21.79 Hertz22.30 Hertz1

Percent Difference (%)

Calculated Frequency (Hertz)

Identified Frequency (Hertz)

Mode Number

(The identified frequencies are simply listed for reference; will be used later to assess the accuracy of the finite element model.)




References• Theory of finite element modeling:

1. Oden, An Introduction to the Mathematical Theory of Finite Elements, Wiley, 1976.

2. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, 1978.

• Implementation and programming:3. Smith, Programming the Finite Element Method, Wiley, 1982.4. Cook, Malkus, Plesha, Concepts and Applications of Finite Element Methods,

Wiley, 1989.

• Applications of the finite element method:5. Bathe, Wilson, Numerical Methods in Finite Element Analysis, Prentice Hall, 1976.6. Zienkiewicz, The Finite Element Method in Engineering Sciences, McGraw-Hill,

1977.7. Dhatt, Touzot, The Finite Element Method Displayed, Wiley, 1984.8. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element

Analysis, Prentice Hall, 1987.

se207: validation of computational...

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