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7/28/2019 Linear Statics Fem Hand Out

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OverviewIntroduction

Linear elasticity problemThe Finite Element Method

Finite Element Method applied to linear statics of

deformable solids

Joel Cugnoni, LMAF / EPFL

February 20, 2013

Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable sol


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1 IntroductionDefinitionsSome MathsLinear elasticity

2

Linear elasticity problemEquilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form

3 The Finite Element MethodApproximate solution of the Weak FormFinite Element Method



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DefinitionsSome MathsLinear elasticity

Some Maths

Symmetric 2nd order tensors written in vector forms:

= {11, 22, 33, 23, 31, 12}T

Differential operator:

=

/x

1 0 00 /x2 00 0 /x30 /x3 /x2

/x3 0 /x1

/x2 /x1 0

hence we have:

div() = T

and:

grad(u) = uJoel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable sol

O i


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Linear elasticity

The stress is defined by the symmetric Cauchy stress tensor

(x) = {11, 22, 33, 23, 31, 12}T

For small deformations, the (infinitesimal) strain tensor(x) = {11, 22, 33, 23, 31, 12}T is defined by:

(x) = u(x)

The constitutive stress - strain relationship of linear elasticity

is given by the Hooks law:

(x) = C(x) (x)

where C(x) is the 4th order elasticity tensor


O i


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Elasticity tensor

For an isotropic material in 3D, the 4th order symmetricelasticity tensor is defined by two constants: the Youngsmodulus E and the Poissons ration

With the previous definitions, C can be written in matrix form:

C =

1 0 0 0 1 0 0 0 1 0 0 0

0 0 0 (12)2 0 0

0 0 0 0 (12)2 0

0 0 0 0 0 (12)2

with: = E(1+)(12)


Overview Equilibrium of a linear elastic solid


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Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form

Stress equilibrium

Internal Stress Equilibrium

Equilibrium is achieved if, atany point x of :

div((x)) + f(x) = 0or

T(x) + f(x) = 0

where:

is the internal Cauchystress tensor

f is the body (volumic)force vector



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Virtual Work Principle

Virtual Work Principle

The weak form of the elastostatic problem can be obtained usingthe Virtual Work Principle. Taking the differential equation, we

multiply it by an arbitrary admissible test function u and integrateit over the domain .

u

TC u + f

d = 0 , u

Note that the initial x of the local equilibrium diff. equation hasbeen replaced by u in the integral form.




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Virtual Work Principle, development..

Applying integration by part, and imposing that u is continuous,differentiable and satisfies the homogeneous form of thedisplacement boundary condition:

u(x) = 0 , x u

We obtain:

(u)T C u d+

uT t d()+

uT fd = 0 , u

which terms corresponds to the virtual work of internal stresses,surface tractions and body forces.



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IntroductionLinear elasticity problem

The Finite Element Method

qDifferential EquationVirtual Work PrincipleWeak Form

Weak form of linear statics

Linear elasticity problem in Weak Form

find the displacement field u(x) U such that :

(u)T

C u d =

uT t d() +

uT f d ,

u V

where:Ui =

ui(x) | ui(x) H

1(); ui(x) = ui(x), x u

Vi =

ui(x) | ui(x) H1(); ui(x) = 0, x u




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Differential EquationVirtual Work PrincipleWeak Form

Comments

The displacement boundary conditions have been moved inthe class of functions U. This type of boundary condition iscalled essential because it is related to the essence of theproblem = the displacement field!!

The surface traction boundary conditions t on are nowexplicitly introduced in the weak form. This type of boundarycondition is called natural because it falls naturally in theweak form of the problem!!

The name Weak Form comes from the fact that therequirements of differentiability / continuity of the fields uand u are relaxed compared to the initial differential equationform. Note that the functions do not necessarilly need to becontinuous!


Overviewf


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Approximate solution of the Weak FormFinite Element Method

Approximate solution of the Weak Form

We are looking for an approximate solution of the Weak Formproblem. We give us two subsets Uh and Vh of the function spacesU and V to form the basis of the approximations:

uh u , uh u

uh

Uh

U , uh

Vh

VBy considering an approximation Uh based on a linear

combination of a finite number of basis functions, we can write:

uh(x) = H(x) q

where H(x) is the 3xn matrix of shape functions and q is thevector of displacement components. Using Galerkin method, wechoose the same base function space for u:

uh(x) = H(x) q



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OverviewI t d ti A i t s l ti f th W k F


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Approximate linear statics problem

Approximate linear statics problem

find the displacement components q such that :

K q = r

and:uh(x) = H(x) q = u , x u

with:

K =

B

T

CB d

r =

HTfd +

HTt d()


OverviewIntroduction Approximate solution of the Weak Form


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Comments

K is called the stiffness matrix of the system, r is the forcevector containing contributions of both surface tractions andbody forcesBy choosing a finite subset of basis functions to build theapproximation of the weak, we have turned the algebraic

differential equation problem into a linear system of equations:all terms of K and r are constant, hence the name linearstatic problem.As the displacement is a linear combination of functions

defined over , enforcing exact displacement boundaryconditions uh(x) = u is not trivial except for zero imposeddisplacement.Integrating the matrices & force vector over an arbitrarydomain can become very complex, and thus numerical

integration methods may be used.Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable sol



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In some sense, the Finite Element Method is one way of defining asystematic approximate function space with the aim to:

1 build approximate solutions over arbitrarily complex domains

2 simplify the treatment of essential boundary conditions3 simplify the integration of the global system matrice(s) and

load vector(s)

4 automate the resolution procedure: integration of the linear

system & solution

Now lets see how this can be achieved !!




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The key ideas behind the Finite Element Method

How to process complex domain?

By discretizing the domain in a set of smaller & simpler domains.

The discretization h

of is defined by a set of discrete pointscalled nodes and a set of sub-domains of basic topology calledelements, which forms what we call a mesh.

h , h = 1 2 ... n

The j-th node is represented by its coordinates xj and the eachelement is defined by its connectivities: an ordered list of nodeswhich serve as support for the element.




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A simple mesh made of 16 elements and 25 nodes




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ppFinite Element Method


An much more complex mesh of an HydroptereJoel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable sol



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Finite Element Method


How to simplify the treatment of essential BC?

Use discrete node-based shape functions on h : each node ofcoordinates xi is associated to its own shape function hi. The

approximation uh

is then written as :

uh = H q with H = [h1 I, h2 I, ..., hn I]

and q =

q1, q2, ..., qp

T

Moreover, the shape functions are required to have compactsupport (Kronecker property):

hi(xj) = ij x

j h




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An illustration of h5(x) showing the compact support property



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Li l ti it blApproximate solution of the Weak FormFi it El t M th d


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How to simplify the integration?

Split integration in each element..

d =n

i=1

i

d

d() =

k

i=1

i d()

Note that local support means that, when integrating hi, onlycontributions of element containing node i must be processed.



Linear elasticity problemApproximate solution of the Weak FormFinite Element Method


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How to simplify the integration?

d =

1

d +

2

d +

3

d +

4

d





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How to automate the process?

By treating the problem per element : localization !Local displacement approximation in the e-th element as :

euh = eH eq

Where:eH = [eh1I,

eh2I, ...,ehpI]

And: eq = eL q

eL is called the binary localization matrix representing the global -local numbering relationship (related to the element connectivitytable)





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An illustration of the global shape function h5(x)



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y pThe Finite Element Method


Integration of the system of equationUsing the localized quantities euh and eH, we can expand theintegration:

K = h

BT C B d =

q

e=1

eLTe

eBT eC eB d eLwhere eB = eH is the local displacement - strain matrixand eC is the element elasticity matrix (strain - stressrelationship).

We then define the elementary stiffness matrix eK:

eK =

e

eBT eC eB d





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y pThe Finite Element Method


Assembly operation

We define the so-called Assembly operator

as:

pe=1

{} =

pe=1

eLT {}e L

With this operator:

K =

pe=1

eLT eKeL =

pe=1

eK





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Automating the integration / standardizing the shapefunctions

To automate the integration and simplify the definition of shapefunctions, we transform each distorted elementary domain e

into a reference domain a

where we can apply standard numericalintegration procedures.

The coordinate transformation

eT : a e | x()

maps any point of coordinate = {1, 2, 3}T in the master

domain a to its (single) corresponding point of coordinatex = x() in the elementary domain e (bijective application).


OverviewIntroductionLinear elasticity problem

Th Fi i El M h d



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An illustration of the coordinate transform eT



Th Fi it El t M th d



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Master element & shape functions

With the coordinate transform eT and eT1, the shape functionsehi(x) can be mapped to master shape functions

ahi() defined inthe master element space a:

eH(x) = eH(x()) = aH()

Such that now, in each element e:

euh[x()] = aH() eq

Thanks to that transformation, we only need to derive once theshape function formulations aH() in the normalized masterelement space a Standardized shape functions !!






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Element & Master shape functions






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How to choose a simple form of coordinate transformation ?

Simply by using the shape functions to write the coordinatetransform:

e

T : x = x() =a

H()e

x

Which means that inside each element e, the coordinates areinterpolated as a linear combination of the shape functions andnodal coordinates ex.

With the Kronecker propertye

hi(xj

) = ij a

hi(j

) = ij, thisensures that each node of the master element corresponds to anode in the deformed element e.






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Coordinate transformation






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Automating the integrationUsing the coordinate transform eT and master shape functions,the integrals can be rewritten to be carried out directly on astandard domain a:

eK =

a

{ eB(x())}T C { eB(x())}

ej da

Where ej is the determinant of the jacobian matrix eJ

e

j = det(

e

J)

with the definition:eJij =

xij



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Numerical integrationOn the master element, standard Gauss-Legendre numericalintegration schemes can be used to automate the calculation of theelementary stiffness matrix eK and load vector. For eachdimension, the integrals are transformed into a weighted sum:

+1

1() dj

ri=1

ij() |j=ij

where the points of coordinates ij are called Gauss points or

integration points and

i

j are the associated weights.Using the integration schemes in the three directions:+11

+11

+11

() d3d2d1 r

i=1

sj=1

tk=1

i1j2

k3 () |1=i1,2=

j2,3=

k3

e = e T e e aJoel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable sol OverviewIntroduction




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Numerical integrationUsing the integration schemes in the three directions:

+11

+11

+11

() d3d2d1 r

i=1

s

j=1

t

k=1

i1j2

k3 () |1=i1,2=

j2,3=

k3

Thus, the elementary stiffness matrix eK can be computednumerically as:

eK

ri=1

sj=1

tk=1

i1j2k3

(aH)T C (aH) ej1=i1,2=

j2,3=

k3

Note that the internal terms (aH)T C (aH) j are evaluateddirectly in the master element coordinate system 1, 2, 3.






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Gauss points & numerical integration






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So finally, what is a Finite Element ?

Using these principles, a standard, displacement based, FiniteElement formulation is defined by:

1 The basic topology of its master element a

2 The coordinate transform eT mapping the master domain a

to the real element domain e

3 The shape functions aH and their derivatives hi/j in themaster element

4 The numerical integration scheme (Gauss points and weights)in the master element.






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So finally, what is a Finite Element Model?

A Finite Element Model is thus defined by :

1 The geometrical mesh h consisting of all elementary domains

e

(defined by nodes and connectivity table)2 The definition of the problem to be solved: K q = r

3 A list of Finite Elements (formulations) supported bygeometrical cells of the mesh

4

Boundary conditions: imposed nodal displacementq on uand imposed surface tractions on

5 Material properties ( eC ) assigned to each Finite Element






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Shape functions

Requirements

1 Continuity: to have a continuous solution, shape functionsmust be continuous along the interfaces between elements

2 Completeness: to be able to represent uniform strain &

displacement, shape functions must at least be affine functions3 Differentiability: shape functions & derivatives must be

sufficiently regular to be square integrable over each element

Consequences

Partition of unity: at each point x,

i hi(x) = 1

Basic choice of shape function: piecewise polynomials, atleast piecewise trilinear functions in 3D


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