Date post: | 14-Apr-2018 |
Category: |
Documents |
Upload: | terence-deng |
View: | 221 times |
Download: | 0 times |
of 31
7/27/2019 FE Review Web
1/31
ransparen
SMA01
Strong and weak
forms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
SMA01: Advanced numerical modeling of
mechanical structuresFinite Element - Basics
D.Brancherie, A. Rassineux
1 / 2 7
http://find/7/27/2019 FE Review Web
2/31
ransparen
SMA01
Strong and weak
forms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Strong and weak forms
Definitions
A scalar problem : the thermal diffusion problemA vectorial problem : the elasticity problem
linear elasticityquasi-static
2 / 2 7
http://find/7/27/2019 FE Review Web
3/31
ransparen
SMA01
Strong and weak
forms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Strong and weak form
Definitions
Strong form :
set of differential or partial differential equations + boundary
conditions written on each point of a domain
D(u(x)) = 0 x
with boundary conditions
BC(u(x)) = 0 x
weak form (or variational formulation) :
an integral version of the strong form
w(x) D(u(x)) = 0 w(x)
with boundary conditions
BC(u(x)) = 0 x
3 / 2 7
http://find/7/27/2019 FE Review Web
4/31
ransparen
SMA01
Strong and weak
formsFinite Elementapproximation
Finite Elementformulation
Implementationissues
A scalar problem : the thermal diffusion
The strong form :
Equilibrium
cTt
= divqth + Pth
for stationary problem :
divqth + Pth = 0 x
Behavior : Fouriers law(homogeneous material)
qth = kT
Boundary conditions
qth = qd ontT = Td onu
qd
t
Tdu
Thermal problem
kT = Pth x
with qth = qd on t and T = Td on u4 / 2 7
http://goforward/http://find/http://goback/7/27/2019 FE Review Web
5/31
ransparen
SMA01
Strong and weak
formsFinite Elementapproximation
Finite Elementformulation
Implementationissues
A sclar problem : the thermal diffusion
Weak form of equilibrium
div qth Pthwd = 0 wuse of the integral theorems :
wkT d
wPth
d
(kT
qthn)w dS = 0 w
remark : weak form and strong form are equivalent
practically, for FEM issues, w can be chosen in KA0
KA0 = {w regular | w = 0 on u}
Weak form for FE issues
find T KA such that
wkT d
wPth
d
(kT
qth
n)w dS = 0 w KA0
where KA = {T regular | T = Td on u}5 / 2 7
http://find/7/27/2019 FE Review Web
6/31
ransparen
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
A vectorial problem : the continuum mechanics problem
The strong form :
Equilibrium (quasi-static)div + b = 0 x
n = t = td on t
Kinematics(small perturbation hypothesis)
(u) = 12
u +uT
u = ud on u
Behavior (linear elasticity, isotropic,homogeneous)
= C : (u)ou = tr((u))1 + 2(u)
td
t
udu
Navier equation for elasticity
( + )(divu) + u + b = 0 x
with u = ud on u and n = t = td on t6 / 2 7
http://find/7/27/2019 FE Review Web
7/31
ransparen
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Continuum mechanics problem
Weak form of equilibrium
(div
+ b) w d = 0 w
use of the integral theorems (Green formula) :
: (w) d
b w d
n
tw dS = 0 w
remark : weak form and strong form are equivalent
practically, for FEM issues, w can be chosen in KA0
KA0 = {w regular | w = 0 on u}
Weak form for FE issues
find u KA such that
: (w) d
b w d
t
td w dS = 0 w KA0
where KA = {u regular | u = ud on u} and = ((u))7 / 2 7
ransparen
http://find/7/27/2019 FE Review Web
8/31
ransparen
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Continuum mechanics problem
Weak form for FE issuesIn the case of linear elastic isotropic homogeneous material, the
solution u solves :
(u) : C : (w) d =
b w d +
t
td w dS w KA0
Remarks :for elastic problems, the solution u can also be defined as theminimum of the potential energy
potential energy :
(v) = 12
(v) : C : (v) d
b v d t
td v dS
u = arg minvKA
(v)
strong /weak form :
derivatives order lower in the weak form than in the strong form8 / 2 7
ransparen
http://find/7/27/2019 FE Review Web
9/31
ransparen
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Summary (for elasticity problem)
Objectivefind u KA solution of :
(u) : C : (w) d =
bw d+
t
tdw dS w KA0
where u = u(x) is a field defined on
9 / 2 7
ransparen
( )
http://find/7/27/2019 FE Review Web
10/31
ransparen
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Summary (for elasticity problem)
Tensor to vector notations (Voigt notations)Second order tensors :
=
xxyyzz
2 xy2 xz2 yz
and =
xxyyzzxyxzyz
Fourth order tensor : elasticity tensor
C =
+ 2 0 0 0 + 2 0 0 0 + 2 0 0 00 0 0 0 00 0 0 0 0
0 0 0 0 0
9 / 2 7
ransparen
S (f )
http://find/7/27/2019 FE Review Web
11/31
ransparen
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Summary (for elasticity problem)
Objectivefind u KA solution of :
(u) : C : (w) d =
bw d+
t
tdw dS w KA0
where u = u(x) is a field defined on
Weak form in Voigt notations
(w)TC(u) d =
wTb d+
t
wTtd dS w KA0
9 / 2 7
ransparen
S (f l i i bl )
http://find/7/27/2019 FE Review Web
12/31
ransparen
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Summary (for elasticity problem)
Objectivefind u KA solution of :
(u) : C : (w) d =
bw d+
t
tdw dS w KA0
where u = u(x) is a field defined on
Weak form in Voigt notations
(w)TC(u) d =
wTb d+
t
wTtd dS w KA0
FE purposefind an approximation uh Sh KA of the solution u solvingthe previous equation by choosing a priori the form of thesolution
the weak form is not solved w KA0 but only forw Vh0 KA0
FE mesh, FE approximation9 / 2 7
ransparen
O li
http://goforward/http://find/http://goback/7/27/2019 FE Review Web
13/31
ransparen
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Outline
Discretisation of the domain : mesh
Approximation, shape functions
10/27
ransparen
M h
http://find/7/27/2019 FE Review Web
14/31
ransparen
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Mesh
Discretisation of the domain into h
Geometry approximation : h approximation of
Mesh :partition of h into elements connected through their nodes
h =
Nele=1
e with e
f(e = f) =
an edgea vertex
node element
1
2 3
Forbidden situation
11/27
ransparen
M h
http://find/7/27/2019 FE Review Web
15/31
ransparen
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Mesh
Practical definition of the meshThe mesh is fully defined by giving
the node coordinate tabledefinition of the node position
node number x y z
1 x1 y1 z1...
......
...i xi yi zi...
......
...
the connectivity tabledefinition of the elements through their nodes
element number node 1 node 2 node 3
1 node 1(1) node 2(1) node 3(1)...
......
... node 1() node 2() node 3()...
.
.....
.
..12/27
ransparen
FE approximation and shape functions
http://find/7/27/2019 FE Review Web
16/31
ransparen
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
FE approximation and shape functions
Main idearepresentation of a field through a minimum number ofparameters (the degrees of freedom)
parameters : the value of the field on the nodes of the mesh
Shape (interpolation) functionsFor each node i of the mesh, a spatial function Ni(x) is definedsuch that :
Ni(xj) = 0 if j = i
1 if j = i
Ni(x)
13/27
ransparen
FE approximation
http://find/7/27/2019 FE Review Web
17/31
p
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
FE approximation
1 2 3 4 5 6 7node
For elasticity, approximated displacement field :
uh(x) =Nnodei=1
Ni(x)di = N(x)d
with di = uh(xi), d the nodal solution vector (displacement,
temperature, ...) andNi(x
) the FE shape functions.
uh(x) =
N1 0 N2 0 ... N Nnode 0
0 N1 0 N2 ... 0 NNnode
N(x)
d1xd1y
.
.
.
dNnodex
dNnodey
d14/27
ransparen
FE approximation
http://find/7/27/2019 FE Review Web
18/31
p
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
FE approximation
1 2 3 4 5 6 7node
Remarks
the field uh(x) is defined for all x through the definition ofd
d is a vector of parameters, of limited length Nnode defining thesolution
for standard FE, shape functions are polynomial functions (moreelaborated for XFEM or isogeometric elements)
15/27
ransparen
Outline
http://find/7/27/2019 FE Review Web
19/31
p
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Outline
FE formulation
16/27
ransparen
From continuum mechanics to FEM
http://find/7/27/2019 FE Review Web
20/31
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
From continuum mechanics to FEM
Construction of the FE problem
Weak form (recall)(u) : C : (w) d =
b w d +
t
td w dS w KA0
FE formulation (Voigt notations)
h
(uh)TC(wh) d =h
wTb d +
t
hwTtd dS wh KA0
FE approximation
solution : uh
(x) =
Nnode
i=1
Ni(x)di = N(x)d
test function : wh(x) =
Nnodei=1
Ni(x)qi = N(x)q
(uh) = Bd where B = LN with L the matrix form ofs
17/27
ransparen
From continuum mechanics to FEM
http://find/7/27/2019 FE Review Web
21/31
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
From continuum mechanics to FEM
FE problemfind uh such that
h
(uh)TC(wh) d =
hwTb d +
th
wTtd dS wh KA0
FE formulation
find d such that q s.t. q = 0 on uh
qT
hBTCB d
d = qT
hNTb d +
th
NTtd dS
18/27
ransparen
FE problem
http://find/7/27/2019 FE Review Web
22/31
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
FE problem
find d such that q s.t. q = 0 on uh
qT
h BTCB dd h N
Tb d th N
Ttd
dS = 0
find d solution of
hBTCB d
K
d = hNTb d +
thNTtd dS
fext
FE problem
find d solution ofKd = fext
Remarks :K : (Ndof Nnode, Ndof Nnode)fext : (Ndof Nnode, 1)
K is a symmetric, def., positive matrix (with a lot of zeros !)19/27
ransparen
Outline
http://find/7/27/2019 FE Review Web
23/31
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Outline
Construction ofK and fext
Reference element
Numerical integration
Numerical resolution
20/27
ransparen
Construction of the FE problem
http://find/7/27/2019 FE Review Web
24/31
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Construction of the FE problem
FE problem - construction ofK and fext
K = hBTCB d =
Neleme=1
eBTCB de
Term ij of matrix K
Kij =Neleme=1
eBTi CBj d
e
case 1 : i and j have disconnected
supports : Kij = 0
case 2 : Supp(i) Supp(j) =
Kij =
eSupp(i)Supp(j)
eBTi CBj d
e
only a limited number of elements have a contribution to KijElement stiffness :
computation on each element ofKe =
eBTeCBe d
Stiffness K obtained by assembly procedure of all Ke
21/27
ransparen
Construction of the FE problem
http://find/7/27/2019 FE Review Web
25/31
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Construction of the FE problem
FE problem - construction ofK and fext
K = hBTCB d =
Neleme=1
eBTCB de
Term ij of matrix K
Kij =Neleme=1
eBTi CBj d
e
case 1 : i and j have disconnected
supports : Kij = 0
case 2 : Supp(i) Supp(j) =
Kij =
eSupp(i)Supp(j)eBTi CBj d
e
only a limited number of elements have a contribution to KijElement stiffness :
computation on each element ofKe =
eBTeCBe d
Stiffness K obtained by assembly procedure of all Ke
21/27
ransparen
Isoparametric finite elements
http://find/7/27/2019 FE Review Web
26/31
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
p
Reference element ( to limit the number of geometry to consider)introduction of the natural coordinates : ,
geometry described by : x =
Nnodei=1
Ni(, )xi
Example for a quadrangular element :
1 2
4 3
1
3
2
4
Ni(, ) =14
(1 i)(1 i)Example for a linear triangle :N1(, ) = 1 , N2(, ) = , N3(, ) =
Isoparametric elementsthe geometry and the solution are interpolated with the sameshape functions
uh =
Nnode
i=1
Ni(, )di and x =Nnode
i=1
Ni(, )xi
22/27
ransparen
Numerical integration
http://find/7/27/2019 FE Review Web
27/31
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
g
Computation ofKe, fe,ext
Numerical integration : only an approximation of the integrals iscomputed :
ref
g(, ) d Nint=1
g(, )w
(, ) : natural coordinates of integration pointsw : integration weight
23/27
ransparen
Numerical integration
http://find/7/27/2019 FE Review Web
28/31
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
g
Computation ofKe (2D case)
Ke = eBTCB g(x,y)
dxdy
1 change of variable : (x, y) (, )
Ke =
eg(x, y) dx dy =
ref
g (x(, ), y(, )) | detJ(, )| d d
2 numerical integration (quadrature formula)
Ke =eg(x, y) dxdy
=
ref
g (x(, ), y(, )) | detJ(, )| d d
Nint
=1
g (x(, ), y(, )) | detJ(, )|w
Ke =
Nint=1
BT(, )CB(, )| det J(, )|w
only the shape functions and their derivatives with respect to
and are needed to implement a finite element24/27
ransparen
Architecture of a FE code
http://find/7/27/2019 FE Review Web
29/31
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Global level of FE code
Equilibrium :
Nelem
Ae=1
fe,int fe,ext
= 0 Ku = fext
Element level
Computation :Ke =
eBTCB dx
fe,int =eBTAdx, fe,ext
Integration points level
Computation : = C
= Bu
uK
e
f
e,int
, fe,ex
25/27
ransparen
Global computation - static
http://find/http://goback/7/27/2019 FE Review Web
30/31
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
General form to be solved :
Ku = fDirect method : (Gauss elimination)
triangular decomposition : K = LUwith L lower triangular part, U upper triangular part(diag(L)=1, diag(U)=diag(K))
solution : Uu = yLy = f
2 lower and upper triangular systems to be solved
Iterative method : conjugate gradient or preconditionnedconjugate gradient
use the terms in K directly, no need to decompose K
efficient for large problems
ifK symmetric definite matrix
26/27
ransparen
Global computation - static
http://find/7/27/2019 FE Review Web
31/31
SMA01
Strong and weakforms
Finite Elementapproximation
Finite Elementformulation
Implementationissues
Gauss elimination : computation cost
system ofn linear algebraic equations
Triangular decomposition : K = LU
additions n3n3
n3multiplications n3n3
divisions n(n1)2Lower triangular system : Ly = f
additions n(n1)2 n2
multiplications n(n1)2
Upper triangular system : Uu = yadditions n(n1)2
n2multiplications n(n1)2divisions n
27/27
http://find/