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Manual of Reference
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Document: R3.01.00
Isoparametric elements
Summary:
This document presents the bases of the isoparametric elements introduced into Code_Aster for
modeling of the continuous mediums 2D and 3D. One first of all recalls the passage of a strong formulation to one
variational formulation, then one details the discretization by finite elements: use of an element of
reference, calculation of the functions of form and evaluation of the elementary terms. One also briefly describes
the principle of the assembly of these terms and the imposition of the boundary conditions, and they are evoked
methods of matric resolution used. Finally the main stages of a calculation by elements are exposed
stop such as it is conceived and established in Code_Aster.
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Contents
1 Principle of the finite element method ............................................................................................ 4
2 Obtaining a variational formulation ............................................................................................ 4
2.1 Strong formulation ............................................................................................................................. 4
2.2 Formulation weak ............................................................................................................................ 5
2.2.1 Functions tests ....................................................................................................................... 5
2.2.2 Formulate of GREEN ............................................................................................................... 5
2.2.3 Variational formulation ...................................................................................................... 6
2.3 Method of resolution ..................................................................................................................... 7
3 Discretization ......................................................................................................................................... 9
3.1 Cutting in finite elements .......................................................................................................... 9
3.2 Choice of the functions tests ................................................................................................................ 9
3.3 Representation of the geometry .................................................................................................... 10
3.3.1 Element of reference ........................................................................................................... 11
3.3.2 Functions of geometrical interpolation ................................................................................. 11
3.3.3 Stamp jacobienne transformation .............................................................................. 12
3.4 Representation of the unknown factors ...................................................................................................... 12
3.4.1 Polynomial base ................................................................................................................. 12
3.4.2 Functions of form .............................................................................................................. 13
3.4.3 Isoparametric element ..................................................................................................... 14
3.4.4 Correspondence between polynomial base and functions of form ......................................... 14
3.5 Calculation of the elementary terms .................................................................................................... 15
3.5.1 Transformation from derived ............................................................................................... the 15
3.5.2 Change of field of integration ................................................................................ 15
3.5.3 Numerical integration: points of GAUSS ........................................................................... 15
3.6 Example ........................................................................................................................................ 16
3.6.1 Variational formulation .................................................................................................... 17
3.6.2 Functions of form .............................................................................................................. 18
3.6.3 Calculation of the elementary terms ........................................................................................... 19
4 matric System ................................................................................................................................ 20
4.1 Assembly of the matrices and vectors elementary ..................................................................... 20
4.2 Imposition of the boundary conditions kinematics ...................................................................... 20
4.3 Resolution of the matric system .................................................................................................... 20
4.4 Estimate of error and improvement of the precision of calculations .................................................... 21
5 Organization of a calculation by finite elements in Code_Aster ........................................................ 21
5.1 Concept of finite element in Code_Aster ..................................................................................... 21
5.2 Initializations of the elements ........................................................................................................... 22
5.3 Calculation of the elementary terms .................................................................................................... 23
5.4 Total resolution ......................................................................................................................... 23
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6 Bibliography ......................................................................................................................................... 23
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1
Principle of the finite element method
The finite element method is employed in many scientific fields to solve
partial derivative equations. It makes it possible to build a simple approximation of
unknown factors to transform these continuous equations into a system of equations of finished size,
that one can write schematically in the form
WITH THE L
=
, where
U
is the vector of the unknown factors,
With
a matrix and
L
a vector.
Initially, one transforms the partial derivative equations (or strong formulation of
problem) in a variational formulation (or weak formulation). The approximate solution is sought
like linear combination of functions given. These functions must be simple but enough
general to be able “well” to approach the solution. They must in particular make it possible to generate one
space of finished size which is as close as one wants space of functions in which
find the solution. From this old idea (method of the balanced residues), various ways
to choose these functions give place to various numerical methods (collocation, methods
spectral, finite elements).
The originality of the finite element method is to take as functions of approximation of
polynomials which are null on almost all the field, and thus take part in calculation only with
particular point neighborhood. Thus, the matrix
With
is very hollow, containing only the terms
of interaction between “close points”, which reduces the calculating time and the place memory necessary to
storage. Moreover, the matrix
With
and the vector
L
can be built by assembly of matrices and
elementary vectors, calculated locally.
2
Obtaining a variational formulation
One can obtain the variational formulation of a problem starting from the equations with the derivative
partial, by multiplying those by functions tests and while integrating by parts. In mechanics of
solids, the weak formulation then obtained is identical to that given by the Principle of Work
Virtual or in certain cases the minimization of the total potential energy of the structure. Let us note
however that for certain problems, the equations of the model are easier to establish in
tally variational (case of the plates and hulls for example).
2.1 Formulation
strong
Let us take an example resulting from the mechanics of the solids. Local equations of static balance of one
structure
, subjected to forces of volume
F
, of imposed displacements
U
D
on part of its
border
D
and of the imposed forces
G
on a part
NR
of its border, are written:
div
.
+ =
=
F
U U
N G
0
=
in
on
on
D
D
NR
where
is the tensor of the stresses and
N
the outgoing normal on the border
. The relations which bind
the tensor of the stresses
with displacements
U
are called relations of behavior.
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The boundary conditions applying to the primal unknown factors (displacements here) are called
boundary conditions of DIRICHLET, or “essential”. Boundary conditions relating to
the forces (or flows in thermics) are called boundary conditions of NEUMANN, or
“natural”.
2.2 Formulation
weak
2.2.1 Functions
tests
That is to say a space
V
functions called functions tests, “sufficiently” regular and cancelling itself on
D
. In mechanics kinematically, this space is called the space of virtual displacements
acceptable. By multiplying the local equations by a function test
v
belonging to space
V
and in
integrating on the field
, one obtains a variational form of the problem, rigorously
equivalent to the form preceding, known as so operational:
{
}
{
}
V
V
D
D
D
D
=
=
=
=
v
v U
v
v
“regular”
on
“regular”
on
,
,
0
To find
such as
on
:
U
v
F v
N G
+
=
=
V
V
D
D
NR
:
(div
)
.
0
éq 2.2.1-1
In what follows, it will be supposed to simplify that the conditions of DIRICHLET are homogeneous,
i.e.
U
D
=
0
; thus, spaces
V
D
and
V
are confused. Processing of the conditions
to the limits of nonhomogeneous DIRICHLET is exposed in the document [R3.03.01].
Note:
One will not discuss in this document functional spaces to which must belong
the functions tests (cf [bib1]).
2.2.2 Formulate of GREEN
The analog of integration by parts for an unspecified field
of border
be called
formulate of GREEN and states itself as follows, in its simplest form:
I
I
D
N D
=
,
where
I
indicate the derivative compared to the direction
I
, and
N
the outgoing normal with the field.
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When
=
Q
, where
Q
is a vector and
a scalar, it is expressed in the following way:
I I
I I
I
I
Q D
Q N D
Q
=
-
D
,
maybe, by using vectorial notations (it
.
indicate the scalar product):
div ()
grad ().
(
)
Q
Q
q.n
+
=
D
D
D
2.2.3 Formulation
variational
By applying the formula of GREEN to the integral [éq 2.2.1-1] and by taking account of the condition with
limits of NEUMANN
ij J
I
N
G
=
on
NR
and of the condition
v
=
0
on
D
, the form is obtained known as
variational of the problem:
To find
such as:
U
v
=
+
V
V
v D
F v D
G v D
D
ij J I
I I
I I
NR
Like the tensor of the stresses
is symmetrical, the equation can be also written:
ij
ij
I I
I I
D
F v D
G v D
NR
() ()
U
v
=
+
,
where
ij
I
J
J I
v
v
()
(
)
v
=
+
1
2
is the tensor of the linearized deformations. One thus finds exactly the expression given by
Principle of Virtual Work in small displacements. The relation between the tensor of the stresses of
CAUCHY
and displacements
U
will be given by a relation of behavior, and is
independent of the writing of the variational formulation (in the elastic case, one has for example
ij
ijkl kl
()
()
U
U
=
).
One of the advantages of the variational formulation is that it integrates all the boundary conditions:
the conditions of DIRICHLET are taken into account in the definition of space
V
functions
tests, while the conditions of NEUMANN appear naturally after integration by parts.
This integration by parts also makes it possible to lower the command of derivation on the unknown factors. Of
more, in a certain number of cases, it symmetrizes the problem in
U
and
v
.
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Two things remain to be made lorque one wrote the variational formulation (or the Principle of Work
Virtual): to take into account the relation of behavior and to set up the algorithmic one
resolution. For this last point, let us give some examples: writing of an algorithm of resolution
of nonlinear system (method of NEWTON for example) for the nonlinear problems,
writing of a diagram of integration in time for the problems of evolution in dynamics (method of
NEWMARK for example)… Consequently, the majority of the variational problems are brought back to find
U
V
D
such as:
(
) =
v
U, v
v
V
has
L
,
()
,
where
has (,)
· ·
is a bilinear, symmetrical form or not, and
L ()
·
a linear form. If
bilinear form is symmetrical and positive, the problem arising is equivalent to a problem of
minimization of a functional calculus, which in static mechanics of the solids is total potential energy
structure.
2.3
Method of resolution
In this document, one presents only the method finite elements in displacements, where
unknown factors are, as its name indicates it, the variables known as primal (displacements in mechanics),
in opposition to the methods in stresses, or the mixed methods. Space
V
is represented by
a discrete space
V
H
. For the methods finite elements in conformity to which us
let us restrict here, space
V
H
is included in
V
: the approximate solution is thus “more rigid” than
exact solution (it over-estimates energy).
It is reminded the meeting that one places oneself here if the boundary conditions of DIRICHLET are
homogeneous. In addition, one confines oneself with the finite elements of LAGRANGE for which
variables are the values of the unknown fields.
For the finite element method of GALERKIN described in this document, the unknown factors and them
functions tests are represented in the same way, by defining a base of functions
{
}
W
I
()
X
of
space
V
H
.
One calls nodes the points of the field where the unknown factors are calculated, and variable nodal or
degrees of freedom scalar unknown factors to the nodes (components of displacement for example).
a many basic functions necessary are equal to the number of nodal variables: for a problem
three-dimensional where the unknown factors are the three components of the vector displacement, the dimension of
the base is three times the number of nodes.
The indices will be used
I J
,
…
to indicate the numbers of the nodes (
NR
on the whole), and indices
,
…
to indicate the numbers of the unknown factors (
M
unknown factors by node). Thus, the vector
discretized displacement
V
H
is written:
U X
X E
H
H
U
()
()
=
, where vectors
E
are the vectors of
base Cartesian. Thereafter, the index will be omitted
H
writings.
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The basic function associated the node
I
for the unknown factor number
will be noted
W
I
()
X
. In this
base
{
}
W
I
I NM
()
X
1
1
, the unknown field is written:
U
U W
I
I
I
NR
()
()
X
X
=
=
1
.
where
U
I
are the nodal variables.
Thus, the problem amounts finding
U
V
D
such as:
(
,
=
=
=
v
E v
v
V
W has
U
L
I
I
M
I
NR
,
)
()
1
1
.
Each choice of
v
allows to obtain an equation. In the finite element method, one takes
like function test
v
successively each function
W
J
E
(when
V
V
D
=
). Thanks to
linearity of
has (,)
· ·
and
L ()
·
, one can write the discrete problem like:
=
=
J
W has
W
U
L W
I
J
I
M
I
NR
J
,
,
(
,
)
(
),
=
E
E
E
1
1
from where the matric system to solve:
WITH THE L
=
,
with:
{
}
With
W has
W
L
L W
U
U
U
U
U
U
U
J I
I
J
J
J
M
I
I
M
I
NR
M
NR
T
=
=
=
(
,
)
(
),
…
.....
…
…
.....
…
,
E
E
E
U
,
and
11
1
1
1
where
NR
is the number of nodes and
M
the number of scalar unknown factors per node (3 for
displacements in 3D).
In fact, one “condenses” the indices two by two: each new index
I
at the same time information contains
on the number of the node
I
and on the local number of the unknown factor
(the condensed index
I
be called it
number of degree of freedom). A term
With
ije
matrix thus contains information on the interaction
between the degrees of freedom
I
and
J
(for example,
I
represent displacement according to
y
with node 12 and
J
displacement according to
Z
with the node 23).
In many cases, the basic functions used for the various unknown factors in a given node
are the same ones:
W
W
I
I
=
. One calls then the common basic function the function associated with
node
I
, and it is noted
W
I
. Subsequently, it will be supposed to simplify the writings that there is not
that only one scalar basic function associated each node.
N.B.:
Let us note that certain authors, of Anglo-Saxon culture for the majority, describe for reasons
histories finite element method like a method of RITZ per pieces.
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3 Discretization
The discretization consists in choosing a base of space
V
H
and to calculate the terms numerically
matrix
With
and of the vector
L
. For that, one expresses the bilinear form
has (,)
· ·
and the linear form
L ()
·
like a sum on elements, defined by basic field division. If one begins again
the mechanical problem presented in the paragraph [§1.2], that gives:
W W has
W
W
L W
F W
G W
I
J
elements
kl
I
kl
J
I
elements
I I
I I
E
E
E
E
E
(,
)
()
(
)
()
=
=
+
Terms
With
ij
, which represents the interaction between two degrees of freedom
I
and
J
are built in
“assembling” the contributions coming from each element which contains the nodes
agents; one proceeds in the same way to build the vector
L
. These contributions,
called elementary terms, are calculated at the time of a loop on the elements and only depend
only variables of the element
E
.
3.1
Cutting in finite elements
The structure is cut out of “pieces” called elements. The data of the co-ordinates of the nodes
elements and connectivities (list of the nodes of each element) constitutes a mesh.
cutting must respect a certain number of rules: in particular, there should not be nor recovery
nor hole.
Let us recall that one calls nodes the points where the unknown factors are calculated. The nodes can be
nodes of the mesh or not (mediums on the sides for example). The number of scalar unknown factors
(or nodal variables) in an element the number of degrees of freedom of the element is called.
3.2
Choice of the functions tests
The functions tests (or functions of the base
{
}
W
I
()
X
) must be dense in space
V
unknown functions, being continuous from one element to another, to allow to calculate the terms simply
elementary
With
ij
and
L
I
, and to generate a matrix
With
hollow and conditioned well. Three first
conditions are met in particular by the choice of polynomial functions. Moreover, to have one
stamp
With
dig, one will make so that the supports of two basic functions associated two
“distant” nodes are disjoined: thus, the corresponding terms of the matrix will be null.
It is reminded the meeting that one places oneself to simplify the writings if only one function is used
basic by node
I
for all the unknown factors. In this case:
W
W
W
I
I
I
=
=
, where
I
is it
condensed number of degree of freedom for the unknown factor
node
I
.
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The choice of the basic functions is then the following: one associates each node
I
a basic function
W
I
who is a polynomial per pieces being cancelled on all the elements not containing the node
I
[Figure 3.2-a]. So
W W has
I
J
(,
)
=
0
if ddl of numbers
I
and
J
are carried by nodes
I
and
J
who do not belong to the same element. One forces moreover this polynomial to be worth 1 with the node
I
,
and
0
in all the other nodes. In other words,
W X
I
J
IJ
(
)
=
. Thus, nodal values of
unknown factors will be the values taken with the nodes by the exact solution:
U X
U
J
J
(
)
=
.
Appear 3.2-a: basic Function associated with a node
In the continuation, one will call function of form associated with the node
I
the trace (or restriction) on the element
considered basic function
W
I
, and it will be noted
NR
I
.
3.3
Representation of the geometry
The calculation of the functions of form for an unspecified element can be rather intricate. In the case
triangles, one can for example use the concept of barycentric co-ordinates of a point by
report/ratio at the three tops. However, in the case of the quadrangles, such a concept is less
current and calculations can be delicate to carry out analytically. This is why one prefers
often to bring back itself to an element known as of reference, simple form, and from which one can generate
all elements of the same family by a geometrical transformation. Functions of form
are then calculated on this noted generic element
R
, and the transport of the sizes on the element
reality
E
is carried out thanks to the knowledge of the geometrical transformation. Let us note however that
the code of thermo hydraulics N3S uses for reasons of performance of the analytical formulas
explicit and not concept of element of reference.
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3.3.1 Element of reference
Let us note
X
punctual coordinates
X
in the absolute reference mark. Points of the element of reference
will be described in terms of co-ordinates known as parametric
. The figure [Figure 3.3.1-a] gives for
a triangular element in 2D the element of reference and the real element. The transformation
must be
bijective and to transform the nodes and sides of the element of reference into nodes and sides of
the real element.
1
2
1
0
1
element of reference
X
2
real element
X
1
Appear geometrical 3.3.1-a: Transformation
3.3.2 Functions of geometrical interpolation
The geometry of the element will be approximate by the means of functions known as of geometrical interpolation:
thus for example, the curved lines of the real element can be represented by segments
on the element of reference.
These noted functions
NR ()
are defined on the element of reference; they make it possible to know them
co-ordinates
X
of an unspecified point of the real element starting from its co-ordinates
of sound
antecedent in the element of reference and the co-ordinates
X
I
nodes (of local number
I
) of
the real element:
X
NR
X
I
I
I
N
=
=
()
1
,
where
N
is the number of nodes of the element, and
I
the number of each node locally to the element.
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3.3.3 Stamp jacobienne transformation
The jacobienne of the transformation is the matrix of the derivative partial of the real co-ordinates
X
compared to the co-ordinates
in the element of reference:
J
X
=
.
By taking account of the definition of the co-ordinates
X
according to the co-ordinates
X
I
nodes,
one obtains an equivalent expression of the matrix jacobienne:
J
NR X
I
I
I
N
=
=
1
,
where
NR
I
are the terms of the tensor
NR
T
, the number of lines is the number of directions
space, and numbers it columns the number of nodes of the element.
Let us note that the tensor
NR
T
depends only on the definition of the element of reference and not on
that of the real element.
The determinant of the matrix jacobienne, useful in calculations which will follow, is called the jacobien
the geometrical transformation. It is nonnull when the transformation
who makes pass from the element of
reference to the real element is bijective, and positive when
respect the orientation of space.
3.4
Representation of the unknown factors
There are two equivalent ways to represent the unknown factors (component displacement in
the mechanical example) in an element: by the coefficients of their polynomial approximation, or by
their nodal values. These two possibilities correspond to the two manners complementary to
to define an element: by the data of a base of students'rag processions, or by the data of the functions of form
associated the nodes. In addition, let us note that an element is known as isoparametric when its functions
of form are identical to its functions of geometrical interpolation. In Code_Aster, all them
finite elements of continuous medium (2D and 3D) are isoparametric.
3.4.1 Base
polynomial
The way simplest to define an element is to choose a polynomial base made up of one
certain number of independent students'rag processions. For a given unknown factor, the number of students'rag processions
used must be equal to the number of nodal variables, i.e. with the number of nodes used for
to represent the unknown factor. In the case of a triangular finite element where one wishes to have them
linear displacements and constant pressure in each element, polynomial bases used
are respectively
{
}
1,
1
2
X X
and
{}
1
. Consequently, one can choose to calculate displacements
with the 3 nodes nodes and the pressure with the central node.
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One generally defines the polynomial base on the element of reference; it contains students'rag processions of
the form
1 2 3
, where
,
and
are positive or null whole exponents. The degree of such
students'rag procession is the entirety
+ +
. The base is known as complete degree
N
when all students'rag processions of
degree
N
are present. In certain cases, one employs incomplete bases. For example, for
quadrangle
Q
1
in 2D, displacements are linear compared to each direction: the base
used is
{
}
1
2
,
1 2 1
. Components
U
1
and
U
2
displacement are thus written:
U
has
has
has
has
U
B
B
B
B
1 1 2
1
2 1
3 2
4 1
2 1 2
1
2 1
3 2
(,)
(,)
2
4 1 2
= +
+
+
= +
+
+
One notes
P
I
()
I
ème
students'rag procession of the base (which includes/understands some
m
). Components of the vector
displacement
()
U
in the element are then given by the formula:
U
p has
I I
I
m
()
()
=
=
1
One will note
the matrix giving the values taken by the students'rag processions of the polynomial base on
nodes of the element of reference:
II
I
I
P
=
()
,
where
I
is the sequence number of the students'rag procession in the base,
I
the number of the node locally to the element and
I
co-ordinates of the node
I
in the element of reference. This matrix is square, its dimension
is the square of the number of nodes of the element.
3.4.2 Functions of form
An equivalent way to define a finite element is to give, for each unknown factor, the expression of
functions of form of the element. For a given scalar unknown factor (component of displacement
according to y for example), there is as much as nodes where the unknown factor must be calculated. In much
case, one uses the same functions of form for all the components of an unknown vector,
but it is not obligatory. In what follows, one will suppose however to simplify the writings
that it is the case.
The functions of form can be defined on the real element
E
: they then are noted
NR
E
()
X
, they
depend on the geometry of the real element, and are thus different from one element to another. It is more
simple to express them on the element of reference, which gives the functions
NR ()
independent of
geometry of the real element. Let us recall that these functions are polynomial on the element, and that
function of form associated with a node given there takes value 1, whereas it cancels in all them
other nodes of the element.
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The unknown factors are expressed then like linear combination of the functions of form, the coefficients
U
I
combination being called nodal variables:
U
NR
U
I
I
I
N
()
()
=
=
1
.
or
()
(
)
U
NR
U
I
I
I
N
()
X
X
=
-
=
1
1
3.4.3 Element
isoparametric
Two types of interpolation thus intervene in the construction of a finite element: the interpolation
geometrical (using the functions
NR ()
) and the interpolation of the unknown factors (using the functions
NR ()
). An element is known as isoparametric when it is based on identical interpolations for its
geometry and its unknown factors:
NR
NR
()
()
=
.
3.4.4 Correspondence between polynomial base and functions of form
There are the relations:
U
p has
I I
I
()
()
=
and
U
NR
U
I
I
I
()
()
=
.
Moreover, it is clear that one a:
U
p has
has
I
I I I
I
m
II I
=
=
=
()
1
. One deduces from it the following relation between
base polynomial and the functions of form:
II
I
I
NR
P
()
()
=
.
Example: triangle
P
1
in 2D
One will note
()
=
1 2
,
parametric co-ordinates in the element of reference.
P
P
P
1
2
1
3
2
1
()
,
()
,
()
,
=
=
=
=
= -
-
-
1 0 0
1 1 0
1 0 1
1 0 0
1 1 0
1 0 1
1
,
,
from where functions of form:
NR
NR
NR
1
1
2
2
1
3
2
1
()
()
()
= - -
=
=
It is checked well that
NR
I
J
IJ
(
)
=
.
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3.5
Calculation of the elementary terms
The elementary terms to calculate are form:
F ((),
(), (),…)
U X
U X
X
U X
X
dx
E
2
2
.
Three types of operations are to be carried out: the transformation of derived compared to
X
in derivative
compared to
, the passage of an integration on the real element with an integration on the element of
reference, and the numerical realization of this integration which is generally made by a formula of
quadrature.
3.5.1 Transformation of the derivative
The transformation of derived is carried out thanks to the matrix jacobienne
J
, according to the rule of
derivation in channel:
U
X
X
U
T
nod
=
=
-
J
NR
U
1
where
U
nod
is the vector of the nodal values of the component
displacement.
The derivative of a higher nature are also obtained by using this rule, even if that gives place
with expressions more complex than we will not clarify here.
3.5.2 Change of field of integration
The passage to integration on the element of reference is carried out by multiplying the intégrande by
determinant of the matrix jacobienne, called jacobien:
F ((),
(), (),…)
F ((),
(), (),…) det (())
U X
U X
X
U X
X
dx
U
U
U
J
D
E
R
=
2
2
2
2
.
3.5.3 Numerical integration: points of GAUSS
In certain particular cases, one can calculate the integrals analytically. For example, for one
triangle in 2D, Jacobien is constant on the triangle, and the intégrandes are brought back to students'rag processions
that one can integrate exactly:
1 2
1
2
0
1
0
1
2
D D
-
=
+ +
! !
(
)!
.
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However, these particular cases are rare, and one prefers to evaluate the integrals numerically in
calling upon formulas of quadrature. Those give an approximation of the integral under
form of a balanced sum of the values of the intégrande in a certain number of points of the element
called points of integration:
G
D
G
R
G
G
G
R
()
(
)
=
1
.
Scalars
G
the weights of integration, and the co-ordinates are called
G
are the co-ordinates
R
points of integration in the element of reference.
In the methods of integration of GAUSS, the points and weights of integration are given of
manner to integrate exactly polynomials of a nature given. It is this type of method which one uses
in Code_Aster then, the points of integration are called points of GAUSS.
Note:
The number of points of selected GAUSS makes it possible to integrate exactly in the element of
reference. In fact, because of the possible non-linearity of the geometrical transformation or
space dependence of the coefficients, integration is not exact in the real element.
However, it is shown that the made error is of a command lower than the error of
discretization induced by the finite element method.
To illustrate the use of the points of GAUSS, let us take as example the case 3D, where one supposes
that one uses
R
1
points in the direction
1
, R
2
in the direction
2
and
R
3
in the direction
3
, that is to say one
total of
R R R R
1 2 3
=
points of GAUSS. It is shown whereas the expression:
G
G
R
I
J K
I
J
K
K
J
I
()
(,
,
)
R
R
R
3
2
1
=
=
=
D
1
2
3
1
1
1
allows to integrate students'rag processions of the type exactly
() () ()
1
2
3
, with
-
-
2
1
2
1
R
,
R
,
1
2
and
-
2
1
R
3
.
3.6 Example
One proposes to detail the calculation of the variational formulation, of the functions of form, of
elementary matrix of thermal rigidity and the elementary vector loading in the case of
the equation of heat (Laplacian) in 2D, for elements of the quadrangle type
Q
1
.
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3.6.1 Formulation
variational
If one calls
K
the coefficient of conduction, and
T
the temperature, the local equations of balance are
:
-
=
=
-
=
div (grad ())
grad
K
T
F
T
K
T
().
in
on
on
0
0
1
N
where
-
K
T
grad ()
is the heat transfer rate and
N
the outgoing normal with the field. One is imposed
temperature
T
0
0
=
on the edge
0
field, and a heat transfer rate
on the edge
1
.
That is to say
the virtual variable associated the temperature. By multiplying the equilibrium equation by
, in
integrating by parts and by holding account of the boundary conditions, the formulation is obtained
variational:
K
T
D
F D
grad () grad ()
=
-
D
1
The elementary terms that one will have to calculate will be thus:
·
the matrix of thermal rigidity elementary:
With
K
NR
NR D
ije
I
J
E
=
grad (
) grad (
)
·
the elementary vector of surface loading:
L
F NR D
I
I
E
=
·
the elementary vector of linear loading:
L
NR D
I
I
E
=
1
In fact, the term corresponding to the linear loading
L
I
is calculated in Code_Aster on one
element of particular edge and not on the edge of the element
E
. The functions of form are thus used
element of edge (which is the traces on the edge of the functions of form of the surface element). It
is thus necessary always to use 2 elements when one wishes to impose a loading or one
boundary condition: an element of “volume” (for
E
) and an element of edge (for
E
).
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3.6.2 Functions of form
One will use quadrangles
Q
1
, where the unknown factors are represented on the polynomial basis
{
}
1
1 2 1 2
,
. The element of reference is the square represented on the figure [Figure 3.6.2-a]:
1
1
2
1
- 1
- 1
Appear Square 3.6.2-a: of reference
One thus has:
=
-
-
-
-
-
-
=
-
-
-
-
-
-
-
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
,
,
and by using the relation
II
I
I
NR
P
()
()
=
, one obtains the expressions of the four functions of form
associated the nodes:
NR
NR
NR
NR
1 1
2
1
2
2 1
2
1
2
3 1
2
1
2
4 1
2
1
2
1
4 1
1
1
4 1
1
1
4 1
1
1
4 1
1
(,
)
(
) (
)
(,
)
(
) (
)
(,
)
(
) (
)
(,
)
(
) (
)
=
-
-
=
+
-
=
+
+
=
-
+
The matrix of derived from the functions of form in the element of reference is:
NR
=
=
- +
-
+
- -
- +
- -
+
-
T
NR
NR
NR
NR
NR
NR
NR
NR
1
2
3
4
1
2
3
4
2
2
2
2
1
1
1
1
1
1
1
1
2
2
2
2
1
4
1
1
1
1
1
1
1
1
,
,
,
,
,
,
,
,
,
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from where the matrix jacobienne
J
who allows to pass from the element of reference to a real element of which
the nodes have the co-ordinates
(
)
X y
I
I
,
, obtained thanks to the relation
J
NR X
I
I
I
=
=
1
4
:
J
=
-
-
+ +
-
-
-
+ +
-
-
-
+ +
-
-
-
+ +
-
1
4
1
1
1
1
1
1
1
1
2
2
1
2
3
4
2
2
1
2
3
4
1
4
1
1
3
2
1
4
1
1
3
2
(
) (
) (
) (
) (
) (
) (
) (
)
(
) (
) (
) (
)
(
) (
) (
) (
)
X
X
X
X
y
y
y
y
X
X
X
X
y
y
y
y
This matrix of command 2 could be calculated at the points of GAUSS when one needs some and
easily reversed.
3.6.3 Calculation of the elementary terms
The elementary matrix
With
K
NR
NR D
ije
I
J
E
=
grad (
) grad (
)
X
X
4 X include/understand 4 = 16 terms, but
as it is symmetrical, only 10 are to be calculated. It is necessary to carry out three operations for
to evaluate each term of the elementary matrix:
·
summon balanced on the points of GAUSS,
·
transformation of the derived ones:
grad (
)
grad (
)
X
J
NR
NR
I
I
=
-
1
,
·
integration on the element of reference by multiplying by the jacobien (determinant of
J
).
One notes here
grad
X
the gradient whose components are the derivative of the functions compared to
co-ordinates
X
, and
grad
the gradient whose components are the derivative of the functions by
report/ratio with the co-ordinates
(they are the columns of the matrix
NR
T
).
One deduces the final expression from it from the elementary term
With
ije
:
With
K
NR
NR
ije
G
I
G
J
G
G
G
NPG
=
-
-
=
J
J
J
1
1
grad (
(
))
grad (
(
)) det ((
))
1
,
where
NPG
indicate the number of points of GAUSS. One of the families of possible points of GAUSS
(because it integrates the elements exactly
Q
1
) for the square of reference
[] []
-
× -
1 1
1 1
,
,
is that where
the points of GAUSS have as co-ordinates
(
,
)
±
±
1
3
1
3
and where the weights of integration are worth 1.
Components of the elementary vector corresponding to the surface loading
L
I
are calculated
in a way even simpler:
L
F
NR
G
G
I
G
G
G
NPG
I
=
=
(
)
(
) det ((
))
J
1
,
where the surface loading
F
is interpolated at the points of GAUSS of parametric co-ordinates
G
.
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4 System
matric
For each element
E
, one knew to calculate the terms known as elementary: elementary matrix
With
E
and
elementary vector
L
E
. The matrix
With
and the vector
L
are obtained by a procedure that one calls
assembly of the elementary terms, described below. One exposes then the principle of
the imposition of the boundary conditions, then one gives a list of methods usable to solve it
matric system obtained. These the last two points are evoked very briefly because they are treated
in other booklets of the reference material [in particular R6].
4.1
Assembly of the matrices and vectors elementary
The assembly consists in deferring the terms
With
ije
and
L
I
E
of each elementary matrix
With
E
and of
each elementary vector
L
E
in the corresponding boxes
With
IJ
and
L
I
matrix
With
and of
vector
L
. The correspondence enters the local numbers
I
and
J
degrees of freedom, and their numbers
total
I
and
J
is given by the table of connectivities belonging to the mesh.
Indeed, the table of connectivities gives, for each element, the absolute numbers of its nodes
(nodes or not). The command in which the nodes of the element are described gives their numbers
buildings in the element of reference (it
K
ième
node described will have the number
K
locally). In addition,
one knows for each node the command of the degrees of freedom: for example, displacement according to
X
,
then displacement according to
y
, then pressure. That makes it possible to number the degrees of freedom
locally in each element. As for the numbers of the degrees of freedom of the total system, they are
obtained after renumerotation of the unknown factors [R2.02.03]. One thus knows, for a given element,
to associate the numbers
I
and
J
local degrees of freedom numbers
I
and
J
degrees of freedom
total.
To carry out the assembly, one carries out a loop on the elements. For each element, one
determine the nodes which it comprises and thus the total numbers of the degrees of freedom considered, and
one adds at the end
With
IJ
the term
With
ije
he corresponding.
4.2
Imposition of the boundary conditions kinematics
Processing of the boundary conditions kinematics of the type
U U
=
D
is described in detail in
booklet [R3.03.01]. They are imposed by a method of duality, by introducing a vector of
multipliers (or parameters) of LAGRANGE
, which leads to the mixed matric system:
WITH B
L
DRUNK U
T
D
+
=
=
4.3
Resolution of the matric system
The preceding linear system can be solved by a certain number of numerical methods.
methods used in Code_Aster are factorization LDL
T
by blocks [R6.02.01], method
multifrontale [R6.02.02], and the packaged combined gradient [R6.01.01].
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4.4
Estimate of error and improvement of the precision of calculations
After having carried out a calculation by finite elements, it is possible to make an allowing postprocessing
to consider the error made: to see on this subject the documents [R4.10.01] and [R4.10.02].
To improve the precision of the results, two tactics are possible:
·
to refine the mesh
·
to use an approximation of a higher nature
- is by increasing the number of nodes of interpolation (family of the elements of
LAGRANGE);
- is by increasing the number of nodal variables, by adding for example them
derived from the unknown factors (family of the elements of HERMIT); this method is not
used in Code_Aster.
5 Organization of a calculation by finite elements in
Code_Aster
One very briefly describes how and in which place the aspects evoked in this document are
established in Code_Aster.
5.1
Concept of finite element in Code_Aster
A type of finite element is defined by:
·
a type of mesh
·
a list of nodes
·
functions of form
·
options of calculation
An element in the mesh is defined by a type of mesh, a geometry (coordinated nodes)
and a topology (ordered list of the nodes). It is the type of modeling chosen in the file of
order which makes it possible to assign to each mesh mesh a type of finite element. The control
AFFE_MODELE
[U4.22.01] assigns to each mesh a type of finite element corresponding to
modeling specified for this mesh. When same modeling is retained for all it
mesh, the use of
AFFE_MODELE
is simple thanks to the use of the key word
ALL
:
“YES”
.
Important remark:
In the contrary case, one should not forget to assign finite elements to the meshs of edge of which
one has need to impose the boundary conditions and loadings, and which one will have taken care of
to create during the manufacture of the mesh.
The operator
AFFE_CHAR_MECA
[U4.25.01], which affects boundary conditions and loadings, goes
also to create finite elements, for example the finite elements which will carry the degrees of freedom
LAGRANGE used in the dualisation of the boundary conditions [R3.03.01].
The operator
AFFE_CARA_ELEM
[U4.24.01] allows to define additional characteristics for
certain types of elements: for example, the thickness of the hulls, orientation of the beams, matrices
of mass and rigidity of the discrete elements.
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An option of calculation indicates the elementary type of calculation that the element is able to calculate. By
example
RIGI_MECA
relate to the calculation of the elementary matrix of mechanical rigidity:
(
) (
)
With
NR
NR
E
ijkl ij
E
kl
E
R
=
()
()
X
X dx
,
The “data” of this option are the geometry
()
R
and the material
()
, supplemented by
temperature if the material depends on it.
The option
CHAR_MECA
refers to the calculation of the elementary vector for a mechanical loading
imposed on the border:
L
gN
E
E
R
=
()
X dx
.
Let us recall that to apply the loadings of border, one uses finite elements of edge
private individuals, and not borders of the finite elements of volume (3D) or surface (2D).
Note:
A developer can sometimes have the choice between creating a new finite element or adding one
option of calculation to an existing element; the choice between these two solutions holds in general
count criteria of data-processing facility (e.g. under-integrated elements).
5.2
Initializations of the elements
The use of elements of reference makes it possible to once carry out a certain number of calculations for
all at the beginning of the execution. These calculations are carried out in the routines
INI
…. called routines
of initialization of the elements. One defines, for each type of element of reference:
·
the number of nodes and their co-ordinates;
·
the number of families of points of GAUSS;
·
the number of points of GAUSS;
·
weights of integration
G
;
·
values of the functions of form at the points of GAUSS
NR
I
G
(
)
;
·
values of derived from the functions of form at the points of GAUSS
NR
I
G
(
)
.
For a given element, one inevitably does not integrate all the elementary terms with the same one
a number of points of GAUSS: for example, one in general uses more points of GAUSS for
stamp of mass that for the matrix of rigidity, because the products of functions of form are degree
higher than the products of their derivative. Another example is the under-integration used in
certain cases. One calls family of points of GAUSS each whole of points of GAUSS
likely to be used.
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Manual of Reference
R3.01 booklet: General references
HI-75/95/023/A
5.3
Calculation of the elementary terms
During the calculation of the elementary terms (in the routines
TE
….), one carries out for each point of
GAUSS following operations:
·
calculation of derived from the functions of form
NR
, X
E
on the real element starting from the co-ordinates
nodes of the element and derivative of the functions of form
NR
,
on the element of
reference;
·
calculation of the matrix jacobienne;
·
recovery of the weight of integration multiplied by Jacobien at the point of GAUSS considered;
·
evaluation of the intégrande (according to the calculated option).
The elementary term is calculated by nap on the points of Gauss while balancing by the weights
of integration.
5.4 Resolution
total
The total resolution takes place in the routines
COp
…. high level corresponding to the controls
user (
MECA_STATIQUE
[U4.31.01],
STAT_NON_LINE
[U4.32.01],
THER_LINEAIRE
[U4.33.02],
etc).
6 Bibliography
[1]
P.G. CIARLET, “The finite element method for elliptic problems”, Studies in Applied
Mathematics, North Holland, 1978.
[2]
R. DAUTRAY, J. - L. LIONS, “Analyzes mathematical and numerical calculation for sciences and
the techniques ", Volume 2, Masson, 1985.
[3]
G. DHATT, G. TOUZOT, “a presentation of the finite element method”, Maloine S.A.,
Paris, 1984.
Code_Aster
®
Version
3
Titrate:
Isoparametric elements
Date:
05/05/95
Author (S):
I. VAUTIER
Key:
R3.01.00-A
Page:
24/24
Manual of Reference
R3.01 booklet: General references
HI-75/95/023/A
Intentionally white left page.