scispace - formally typeset

Journal ArticleDOI

A new hybrid explicit/implicit in-plane-out-of-plane separated representation for the solution of dynamic problems defined in plate-like domains

01 Nov 2018-Computers & Structures (Pergamon)-Vol. 210, pp 135-144

TL;DR: A new efficient hybrid explicit/implicit in-plane-out-of-plane separated representation for dynamic problems defined in plate-like domains that allows computing 3D solutions with the stability constraint exclusively determined by the coarser in-planes discretization.
Abstract: The present paper extends in-plane-out-of-plane separated representations successfully used for addressing fully 3D model solutions defined in plate-like domain, to dynamics. Common time integration are performed using explicit or implicit strategies. Even if the implementation of implicit integration schemes into a 3D in-plane-out-of-plane separated representation does not imply major difficulties, the use of explicit integration preferable in many applications becomes a tricky issue. In fact the mesh employed for discretizing the out-of-plane dimension (thickness) determines the maximum time-step ensuring stability. In this paper we introduce a new efficient hybrid explicit/implicit in-plane-out-of-plane separated representation for dynamic problems defined in plate-like domains that allows computing 3D solutions with the stability constraint exclusively determined by the coarser in-plane discretization.
Topics: Discretization (51%), Dynamic problem (50%)

Content maybe subject to copyright    Report

HAL Id: hal-02162160
https://hal.archives-ouvertes.fr/hal-02162160
Submitted on 21 Jun 2019
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
A new hybrid explicit/implicit in-plane-out-of-plane
separated representation for the solution of dynamic
problems dened in plate-like domains
Giacomo Quaranta, Brice Bognet, Rubén Ibáñez, Alain Trameçon, Eberhard
Haug, Francisco Chinesta
To cite this version:
Giacomo Quaranta, Brice Bognet, Rubén Ibáñez, Alain Trameçon, Eberhard Haug, et al.. A new
hybrid explicit/implicit in-plane-out-of-plane separated representation for the solution of dynamic
problems dened in plate-like domains. Computers and Structures, Elsevier, 2018, 210, pp.135-144.
�10.1016/j.compstruc.2018.05.001�. �hal-02162160�

A new hybrid explicit/implicit in-plane-out-of-plane separated
representation for the solution of dynamic problems defined in
plate-like domains
G. Quaranta
a
, B. Bognet
a
, R. Ibañez
a
, A. Tramecon
b
, E. Haug
b
, F. Chinesta
c,
a
ESI GROUP Chair & High Performance Computing Institute, Ecole Centrale Nantes, 1 rue de la Noe, BP 92101, F-44321 Nantes cedex 3, France
b
ESI GROUP, 99 rue des Solets, 94513 Rungis, France
c
ESI GROUP Chair & PIMM Laboratory, ENSAM ParisTech, 151 Boulevard de l’Hôpital, F-75013 Paris, France
article info
Article history:
Received 28 October 2017
Accepted 1 May 2018
Available online 8 September 2018
Keywords:
Dynamics
Implicit
Explicit
Hybrid time discretization
Plates
Laminates
PGD
Separated representation
abstract
The present paper extends in-plane-out-of-plane separated representations successfully used for
addressing fully 3D model solutions defined in plate-like domain, to dynamics. Common time integration
are performed using explicit or implicit strategies. Even if the implementation of implicit integration
schemes into a 3D in-plane-out-of-plane separated representation does not imply major difficulties,
the use of explicit integration preferable in many applications becomes a tricky issue. In fact the mesh
employed for discretizing the out-of-plane dimension (thickness) determines the maximum time-step
ensuring stability. In this paper we introduce a new efficient hybrid explicit/implicit in-plane-out-of-
plane separated representation for dynamic problems defined in plate-like domains that allows comput-
ing 3D solutions with the stability constraint exclusively determined by the coarser in-plane
discretization.
1. Introduction
Many mechanical systems and complex structures involve plate
and shell parts whose main particularity is having a characteristic
dimension (the one related to the thickness) much lower that the
other ones (in-plane dimensions). The introduction of appropriate
kinematic and mechanic hypotheses allow the reduction of the
general 3D mechanical problem to a 2D involving the in-plane
coordinates. This was the route employed for deriving beam, plate
and shell theories in solid mechanics, that were extended later to
many other physics, like flows in narrow gaps, thermal or electro-
magnetic problems in laminates, among many others. However, in
many cases, when addressing complex coupled physics the validity
of hypotheses able to reduce models from 3D to 2D becomes
doubtful and consequently in order to ensure accurate Results 3D
discretizations seem compulsory. However the last imply too fine
meshes when considering well-experienced mesh-based dis-
cretization procedures, where the mesh size is almost determined
by the domain thickness and the material and/or solution details to
be represented. In order to alleviate the associate computational
complexity authors proposed few years ago computing the fully
3D solution employing an in-plane-out-of-plane separated repre-
sentation whose computational complexity remains the one char-
acteristic of 2D plate or shell simulations [5,6].
In many structural analysis and simulation of forming processes
dynamical aspects cannot be neglected and then elastic models are
replaced by their elastodynamics counterparts. It exists a vast liter-
ature on structural dynamics, covering different discretization
techniques and time integration procedures [29,17,27,30,26].
When considering an implicit analysis, solution at each time
step needs some iterations to enforce equilibrium. On the contrary
explicit schemes do not require iteration as the nodal accelerations
are solved directly, and from which velocities and displacements
are calculated by simple integration. At its turn displacements
allow the calculation of strains and stresses. The main handicap
of explicit simulations is that the time step must verify the stability
condition, decreasing with the element size.
On the contrary implicit elastodynamic integrations become
unconditionally stable, that is, there is not a limit in the time step
to be considered in what concerns stability. Thus, implicit time
steps are generally several orders of magnitude larger than the
Corresponding author.
E-mail addresses: Giacomo.Quaranta@eleves.ec-nantes.fr (G. Quaranta), brice.
bognet@ec-nantes.fr (B. Bognet), Ruben.Ibanez-Pinillo@eleves.ec-nantes.fr
(R. Ibañez), Alain.Tramecon@esi-group.com (A. Tramecon), Eberhard.Haug@
esi-group.com (E. Haug), Francisco.CHINESTA@ensam.edu (F. Chinesta).

ones considered in explicit time integrations. However, implicit
integration requires the solution of linear systems several times
at each loading step when addressing nonlinear models. Explicit
techniques do not require that matrix inversion and consequently
address nonlinearities (contact or material nonlinearities) easily. In
[18] a hybrid schema was proposed that considers the domain
composed of two parts in which explicit and implicit time integra-
tions apply.
When dynamics applies on degenerated domains, like plates or
shells, and no acceptable simplifying hypotheses are available for
reducing their complexity to 2D, fully 3D solutions seem compul-
sory. This is for example the case when considering the progressive
dynamic damage of composite laminates where a rich through-
the-thickness description could be extremely valuable, among
many other scenarios in which a fully 3D formulation is retained.
In plane-out-of-plane separated representations, revisited in
the next section, allows reducing the 3D solution to a sequence
of 2D (in-plane) and 1D (along the thickness) problems, as proved
when considering elastostatics in plate and shell domains [5,6].
Even if the implementation of implicit integration schemes into
a 3D in-plane-out-of-plane separated representation does not
imply major difficulties, the use of explicit integration, preferable
in many applications, e.g. crash simulations, becomes a tricky
issue. In fact the mesh employed for discretizing the out-of-plane
dimension (thickness) determines the limit time-step ensuring sta-
bility, and consequently it could become quickly unaffordable
when refining the out-of-plane discretization.
In this paper we introduce a new hybrid explicit/implicit in-
plane-out-of-plane separated representation for dynamic prob-
lems defined in plate-like domains that computes efficiently 3D
solution and where the stability constraints are exclusively deter-
mined by the coarser in-plane discretizations.
Next section revisits the main concepts related to the separated
representations intensively considered in the present work. Then,
Section 3 defines the elastodynamics pr oblem and its in-plane-out-
of-plane separated representation. Section 4 addr esses time integra-
tion within the separated representation framework, and proposes
an efficient hybrid explicit/implicit formulation. Finally Section 5
validates the proposed methodology from som e case studies.
2. An overview on separated representations
Separated representations, at the heart of the so-called Proper
Generalized Decomposition PGD consists of expressing the
unknown field depending on many coordinates (space, time,
parameters, ...) as a finite sum decomposition in which each term
involved in the sum consists at its turn in the product of a series of
unknown functions each one depending on one coordinate. Thus,
when addressing a transient model involving the unknown field
uðx; tÞ, its separated representation reads [20–22]
uðx; tÞ
X
N
i¼1
X
i
ðxÞT
i
ðtÞ; ð1Þ
where neither the time-dependent functions T
i
ðtÞ nor the space
functions X
i
ðxÞ are ‘‘a priori” known. Both will be computed on-
the-flight when solving the problem.
This rationale can be extended to the solution of any problem
whose solution involves d generic coordinates uðx
1
; ; x
d
Þ accord-
ing to [1–3]
uðx
1
; x
2
; ; x
d
Þ
X
N
i¼1
X
1
i
ðx
1
ÞX
2
i
ðx
1
ÞX
d
i
ðx
d
Þ; ð2Þ
where the set of coordinates can include a series of parameters
uðx; t; p
1
; ; p
p
Þ according to [8]
uðx; t; p
1
; ; p
p
Þ
X
N
i¼1
X
i
ðxÞT
i
ðtÞ
Y
p
k¼1
P
k
i
ðp
k
Þ: ð3Þ
In the present paper we are mainly concerned by the space sep-
aration to address the solution of mechanical problems defined in
degenerated domain. Next section revisits space separated
representations.
Sometimes the spatial domain
X
, assumed three-dimensional,
can be fully or partially separated, and consequently it can be
expressed as
X
¼
X
x
X
y
X
z
or
X
¼
X
xy
X
z
respectively. The
first decomposition is related to hexahedral domains whereas the
second one is related to plate, beams or extruded domains. We
consider below both scenarii:
The spatial domain
X
is partially separable. In this case the sep-
arated representation writes:
uðx; zÞ
X
N
i¼1
X
i
ðxÞZ
i
ðzÞ; ð4Þ
where x ¼ðx; yÞ2X
xy
and z 2 X
z
. This decomposition implies:
(1) the solution in
X
xy
of two-dimensional BVP’s to obtain func-
tions X
i
,
(2) the solution in
X
z
of one-dimensional BVP’s to obtain func-
tions Z
i
.
The complexity of the PGD simulation scales with the two-
dimensional mesh used to solve the BVP’s in
X
xy
, regardless of
the mesh used in the solution of the BVP defined in
X
z
for calculat-
ing Z
i
ðzÞ.
The spatial domain
X
is fully separable. In this case the sepa-
rated representation writes:
uðx; y; zÞ¼
X
N
i¼1
X
i
ðxÞY
i
ðyÞZ
i
ðzÞ; ð5Þ
implying:
(1) the solution in
X
x
of one-dimensional BVP’s to obtain func-
tions X
i
,
(2) the solution in
X
y
of one-dimensional BVP’s to obtain func-
tions Y
i
,
(3) the solution in
X
z
of one-dimensional BVP’s to obtain func-
tions Z
i
.
The cost savings provided by the PGD are potentially phenomenal
when the spatial domain is fully separable. Indeed, the complexity
of the PGD simulation now scales with the one-dimensional
meshes used to solve the BVP’s in
X
x
;
X
y
and
X
z
.
Even when the domain is not fully separable, a fully separated
representation could be considered by using appropriate geometri-
cal mappings or by immersing the non-separable domain into a
fully separable one. The interested reader can refer to [16,13].
In-plane-out-of-plane separated representations are particu-
larly useful for addressing the solution of problems defined in plate
[5], shell [6], beams [7] or extruded domains [23]. The same
approach was extensively considered in structural plate and shell
models in [11,31–35,28]. Space separated representations where
enriched with discontinuous functions for representing cracks in
[15], delamination in [24] and thermal contact resistances in
[10]. Domain decomposition within the separated space represen-
tation was accomplished in [25]. The in-plane-out-of-plane
decomposition was then extended to many other physics. Thermal
models were considered in [10] and squeeze flows of Newtonian
and non Newtonian fluids in laminates in [12,14,19].
As soon as separated representations are considered the solu-
tion of a multidimensional problem reduces to the solution of a
sequence of lower dimensional problems with the consequent

computing time savings. The solution procedure has been exten-
sively used, described and analyzed in our former works, many
of them referred in the present work. The interested reader can
refer to the primer [9] and the numerous references therein.
3. Elastodynamics: problem definition
We consider a physical domain
X
for which a linear elastic
behavior is assumed, according to
r
¼ C :
; ð6Þ
where C is the fourth order stiffness tensor, and the strain tensor
derives from the symmetric component of the gradient of displace-
ments i.e.
¼ r
s
u, where r
s
refers to the symmetric component.
From now on we consider Voigt notation, and for the sake of
notational simplicity we consider the same notation,
r
;
and C
for expressing the stress and strain vectors and the stiffness matrix
respectively.
The dynamic problem, in absence of damping and external
forces, with the displacement field uðx; tÞ for x 2
X
and
t 2 I ¼½0; T, reads
q
uðx; tÞ¼
r
r
; ð7Þ
with q the material density,
_
u and
u the first and second time
derivative of the displacement field respectively, i.e. the velocity
and acceleration.
The domain boundary C ¼ @
X
is partitioned in the so-called
Dirichlet and Neumann regions,
C
D
and C
N
, where respectively dis-
placements and tractions are enforced, with
C
D
[ C
N
¼ C and
C
D
\ C
N
¼ £. Dynamic problems require specifying the initial
displacement and velocity that without loss of generality in what
follows are assumed null, i.e.
_
uðx; t ¼ 0Þ¼0 and uðx; t ¼ 0Þ¼0.
Assuming again the trial and test displacements belonging to
appropriate functional spaces, and considering an elastic constitu-
tive equation, the weak form associated with (7) reads
q
Z
X
u
u dx þ
Z
X
ðu
Þ C
ðuÞ
ðÞ
dx ¼
Z
C
N
u
F dx; ð8Þ
where the applied traction depends on time, i.e. F ¼ FðtÞ.
3.1. In-plane-out-of-plane separated representation
As discussed in the previous section, with
X
having one dimen-
sion (the one related to the thickness) much smaller than the
others involving the in-plane coordinates, an in-plane-out-of-
plane separated representation seems again the most appealing
route for addressing 3D discretizations while keeping the compu-
tational complexity the one characteristic of 2D discretizations.
The domain is expressed from
X
¼
X
xy
X
z
.
Even if as also indicated space-time separated discretizations
were considered many times in the past [20,2], in the present work
time derivatives are discretized using standard schemes.
By considering the notation uðx; y; z; t ¼ k
D
tÞ¼u
k
ðx; y; zÞ,
with
D
t the time step, the in-plane-out-of-plane separated repre-
sentation of the displacement field at time t
k
¼ k
D
t; u
k
ðx; y; zÞ,
reads
u
k
ðx; y; zÞ¼
u
k
ðx; y; zÞ
v
k
ðx; y; zÞ
w
k
ðx; y; zÞ
0
@
1
A
u
k
N
ðx; y; zÞ
¼
X
N
i¼1
u
i;k
xy
ðx; yÞu
i;k
z
ðzÞ
v
i;k
xy
ðx; yÞ
v
i;k
z
ðzÞ
w
i;k
xy
ðx; yÞw
i;k
z
ðzÞ
0
B
@
1
C
A
¼
X
N
i¼1
U
i;k
xy
ðx; yÞU
i;k
z
ðzÞð9Þ
where ‘‘ refers to the Hadamard product, and with
U
i;k
xy
ðx; yÞ¼
u
i;k
xy
ðx; yÞ
v
i;k
xy
ðx; yÞ
w
i;k
xy
ðx; yÞ
0
B
B
@
1
C
C
A
¼
u
i;k
xy
v
i;k
xy
w
i;k
xy
0
B
B
@
1
C
C
A
; ð10Þ
U
i;k
z
ðzÞ¼
u
i;k
z
ðzÞ
v
i;k
z
ðzÞ
w
i;k
z
ðzÞ
0
B
@
1
C
A
¼
u
i;k
z
v
i;k
z
w
i;k
z
0
B
@
1
C
A
; ð11Þ
where for alleviating the notation the coordinate dependences will
be omitted.
From all them we can obtain the separated vector form of the
strain tensor at time t
k
;
k
ðu
k
Þ:
ðu
k
Þ
X
N
i¼1
@u
i;k
xy
@x
u
i;k
z
@
v
i;k
xy
@y
v
i;k
z
w
i;k
xy
@w
i;k
z
@z
@u
i;k
xy
@y
u
i;k
z
þ
@
v
i;k
xy
@x
v
i;k
z
@w
i;k
xy
@x
w
i;k
z
þ u
i;k
xy
@u
i;k
z
@z
@w
i;k
xy
@y
w
i;k
z
þ
v
i;k
xy
@
v
i;k
z
@z
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
: ð12Þ
The separated representation constructor proceeds by comput-
ing a term of the sum at each iteration. Assuming that the first
n 1 modes (terms of the finite sum) of the solution were already
computed, u
k
n1
ðx; y; zÞ with n P 1, the solution enrichment reads:
u
k
n
ðx; y; zÞ¼u
k
n1
ðx; y; zÞþU
n;k
xy
ðx; yÞU
n;k
z
ðzÞð13Þ
where both vectors U
n;k
xy
and U
n;k
z
are unknown at the present itera-
tion defining a nonlinear problem. The test function u
reads
u
¼ U
xy
U
n;k
z
þ U
n;k
xy
U
z
.
With both U
n;k
xy
and U
n;k
z
unknown the resulting problem
becomes non-linear. We proceed by considering the simplest lin-
earization strategy, an alternated directions fixed point algorithm
widely considered and described in our former works.
When U
n;k
z
is assumed known, we consider the test function u
H
given by U
H
xy
U
n;k
z
. By introducing the trial and test functions into
the weak form and then integrating in
X
z
because all the functions
depending on the thickness coordinate are known, we obtain a 2D weak
formulation defined in
X
xy
whose discretization (by using a standard
discretization strategy, e.g. finite elements) allows computing U
n;k
xy
.
Analogously, when U
n;k
xy
is assumed known, the test function u
H
is given by U
n;k
xy
U
H
z
. By introducing the trial and test functions
into the weak form and then integrating in
X
xy
because all the
functions depending on the in-plane coordinates ðx; yÞ are at pre-
sent known, we obtain a 1D weak formulation defined in
X
z
whose
discretization (using any technique for solving standard ODE equa-
tions) allows computing U
n;k
z
.
Thus, the 3D computational cost is transformed into a sequence of
2D and 1D solutions, with the associated computing time savings [5].
4. Time discretization
Before introducing the hybrid strategy we consider at time t
kþ1
the standard implicit and explicit formulations (two commun time
integration schemas among other possibilities [4]), given respec-
tively by
q
Z
X
u
u
kþ1
2u
k
þ u
k1
D
t
2
dx þ
Z
X
ðu
Þ C
u
kþ1
þ u
k1
2

dx
¼
Z
C
N
u
F
kþ1
þ F
k1
2
dx; ð14Þ

that as previously indicated is unconditionally stable, and the expli-
cit one
q
Z
X
u
u
kþ1
2u
k
þ u
k1
D
t
2
dx þ
Z
X
ðu
Þ C
ðu
k
Þ

dx
¼
Z
C
N
u
F
k
dx; ð15Þ
that is conditionally stable, with the stability limit
D
t
max
, defining
the stability domain
D
t <
D
t
max
, given by
D
t
max
¼
L
c
; ð16Þ
where L is the characteristic length of the spatial discretization and
the dilatational wave speed c is given by
c ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E 1
m
ðÞ
1 þ
m
ðÞ1 2
m
ðÞ
q
s
: ð17Þ
As previously commented explicit strategies are employed in
many commercial codes. However, when applied to discretize 3D
problems defined in degenerated domains, like plates or shells,
the extremely fine meshes considered along the thickness direc-
tion have an unfavorable impact on the time step that becomes
extremely small to ensure stability. The in-plane-out-of-plane sep-
arated representation cannot scape to this important issue, being
the mesh size along the out-of-plane coordinate (much finer that
the one used in the plane) the one that determines the time step.
It is important emphasizing the main aim of the present work
and the proposed methodology for performing it. First, it is impor-
tant to note that we are interested in performing fully 3D simula-
tions in degenerated geometries (e.g. plate domains) while
considering explicit time integrations.
In this context the following remarks can be addressed:
When using 2D discrete models (considering for example plate
elements), the stability criterion related to explicit time integra-
tions involves the size of the elements, but as the mesh is the
one related to the middle plane, the critical time step remains
reasonable in most of cases.
However, as soon as 3D discretizations are considered, the char-
acteristic size of the finite elements along the plate thickness
becomes much smaller than the in-plane characteristic length,
and then when considering explicit time integrations the time
step needed for ensuring stability decreases with the through-
of-thickness characteristic element length.
Increasing the resolution in the thickness direction implies the
increase of the number of elements involved in the discretization
as well as the decrease of the time step for ensuring stability, both
having unfavorable consequences on the computational cost.
In our former works [5,6] we proposed in the framework of elasto-
statics considering in-plane-out-of-plane separated representations
that allowed reducing the computational complexity of solving a
fully 3D problem to the one characteristic of 2D solutions.
However, as just indicated, such a decomposition when combined
with explicit time integrations fails, because again the stability is
associated to the smallest discretization characteristic length, the
one related to the through-of-thickness discretization.
It is in that impasse that one is tempted of using, in the case of
explicit time integration, the in-plane-out-of-plane separated
representation (that reduces the computational complexity to
the one characteristic of 2D models) combined with an hybrid
time integration, explicit in the plane (conditionally stable but
with the critical time-step scaling with the characteristic in-
plane discretization length) and implicit along the thickness
(unconditionally stable), that allows reducing the computa-
tional complexity while keeping as stability constraint the one
associated to the in-plane explicit time integration.
Obviously fully implicit in-plane-out-of-plane decompositions are
possible, where the implicit time integration ensures uncondi-
tional stability while the space separated representation reduces
the computational complexity. Despite of its intrinsic interest it
is not considered in the present paper, and in all cases, the asso-
ciated solutions are the same as the ones obtained by using a fully
3D finite element discretization but reducing the solution compu-
tational complexity. As previously commented fully explicit inte-
grations fail because the too stringent stability conditions induced
by the too fine through-of-thickness discretization.
Thus, in this paper we analyze the intermediate procedure, the
one in which the fine through-of-thickness representation is alle-
viated thanks to the use of the in-plane-out-of-plane space sepa-
rated representation and its associated implicit unconditionally
stable time integration. Thus, the stability of the resulting dis-
cretization is expecte d being induced by the in-plane mesh in
which an explicit time integration is retained. The present paper
is intended analyzing this hybrid methodology, and proving that
in the case of fully explicit separated representations (as in the
case of fully explicit 3D finite elements) the stability is dictated
by the smallest characteristic discretization length (the one along
the domain thickness). On the contrary when considering the
hyb
rid scheme described in the next section, we expect the stabil-
ity being dictated by the characteristic in-plane discretization
length (being the trough-of-thinness discretization implicit).
In summary, the main goal is enriching explicit 2D plate and
shell formulations widely employed in industry and commercial
codes, with a fine through-of-thickness description (3D) without
affecting unfavorably the integration stability.
4.1. Explicit-in-plane/implicit-out-of-plane hybrid scheme
As just indicated, in order to circumvent the just referred stabil-
ity issues, we propose an out-of-plane implicit discretization
(unconditionally stable) while the in-plane discretization (imply-
ing coarser meshes) makes use of an explicit schema. Thus, the sta-
bility is prescribed by the in-plane size mesh, several order of
magnitude higher than the one associated to the thickness.
For that purpose we propose considering at time t
k
the strain
defined by
h
ðu
k
Þ¼
u
k
;x
v
k
;y
w
kþ1
;z
þw
k1
;z
2
v
kþ1
;z
þ
v
k1
;z
2
þ w
k
;y
u
kþ1
;z
þu
k1
;z
2
þ w
k
;x
u
k
;y
þ
v
k
;x
0
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
A
X
N
k
i¼1
u
i;k
xy;x
u
i;k
z
X
N
k
i¼1
v
i;k
xy;y
v
i;k
z
P
N
kþ1
i¼1
w
i;kþ1
xy
w
i;kþ1
z;z
þ
P
N
k1
i¼1
w
i;k1
xy
w
i;k1
z;z
2
P
N
kþ1
i¼1
v
i;kþ1
xy
v
i;kþ1
z;z
þ
P
N
k1
i¼1
v
i;k1
xy
v
i;k1
z;z
2
þ
X
N
k
i¼1
w
i;k
xy;y
w
i;k
z
P
N
kþ1
i¼1
u
i;kþ1
xy
u
i;kþ1
z;z
þ
P
N
k1
i¼1
u
i;k1
xy
u
i;k1
z;z
2
þ
X
N
k
i¼1
w
i;k
xy;x
w
i;k
z
X
N
k
i¼1
u
i;k
xy;y
u
i;k
z
þ
X
N
k
i¼1
v
i;k
xy;x
v
i;k
z
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
ð18Þ

Citations
More filters

Journal ArticleDOI
TL;DR: An enrichment procedure able to address 3D local behaviors, preserving the direct minimally-invasive coupling with existing plate and shell discretizations is proposed and will be extended to inelastic behaviors and structural dynamics.
Abstract: Most of mechanical systems and complex structures exhibit plate and shell components. Therefore, 2D simulation, based on plate and shell theory, appears as an appealing choice in structural analysis as it allows reducing the computational complexity. Nevertheless, this 2D framework fails for capturing rich physics compromising the usual hypotheses considered when deriving standard plate and shell theories. To circumvent, or at least alleviate this issue, authors proposed in their former works an in-plane-out-of-plane separated representation able to capture rich 3D behaviors while keeping the computational complexity of 2D simulations. However, that procedure it was revealed to be too intrusive for being introduced into existing commercial softwares. Moreover, experience indicated that such enriched descriptions are only compulsory locally, in some regions or structure components. In the present paper we propose an enrichment procedure able to address 3D local behaviors, preserving the direct minimally-invasive coupling with existing plate and shell discretizations. The proposed strategy will be extended to inelastic behaviors and structural dynamics.

3 citations


Journal ArticleDOI
TL;DR: This paper proposes an efficient integration of fully 3D descriptions into existing plate software to capture rich 3D behaviors while keeping the computational complexity the one of 2D simulations.
Abstract: Most of mechanical systems and complex structures exhibit plate and shell components. Therefore, 2D simulation, based on plate and shell theory, appears as an appealing choice in structural analysis as it allows reducing the computational complexity. Nevertheless, this 2D framework fails for capturing rich physics compromising the usual hypotheses considered when deriving standard plate and shell theories. To circumvent, or at least alleviate this issue, authors proposed in their former works an in-plane–out-of-plane separated representation able to capture rich 3D behaviors while keeping the computational complexity the one of 2D simulations. In the present paper we propose an efficient integration of fully 3D descriptions into existing plate software.

3 citations


Journal ArticleDOI
TL;DR: This work proposes a space separation with a time adaptive number of modes to efficiently capture transient wave propagation in separable domains and shows that the PGD solution approximates its standard finite element solution counterpart with acceptable accuracy, while reducing the storage needs and the computation time.
Abstract: Transient wave propagation problems may involve rich discretizations, both in space and in time, leading to computationally expensive simulations, even for simple spatial domains. The Proper Generalized Decomposition (PGD) is an attractive model order reduction technique to address this issue, especially when the spatial domain is separable. In this work, we propose a space separation with a time adaptive number of modes to efficiently capture transient wave propagation in separable domains. We combine standard time integration schemes with this original space separated representation for empowering standard procedures. The numerical behavior of the proposed method is explored through several 2D wave propagation problems involving radial waves, propagation on long time analyses, and wave conversions. We show that the PGD solution approximates its standard finite element solution counterpart with acceptable accuracy, while reducing the storage needs and the computation time (CPU time). Numerical results show that the CPU time per time step linearly increases when refining the mesh, even with implicit time integration schemes, which is not the case with standard procedures.

Journal ArticleDOI
Abstract: The use of mesh-based numerical methods for a 3D elasticity solution of thick plates involves high computational costs. This particularly limits parametric studies and material distribution design problems because they need a large number of independent simulations to evaluate the effects of material distribution and optimization. In this context, in the current work, the Proper Generalized Decomposition (PGD) technique is adopted to overcome this difficulty and solve the 3D elasticity problems in a high-dimensional parametric space. PGD is an a priori model order reduction technique that reduces the solution of 3D partial differential equations into a set of 1D ordinary differential equations, which can be solved easily. Moreover, PGD makes it possible to perform parametric solutions in a unified and efficient manner. In the present work, some examples of a parametric elasticity solution and material distribution design of multi-directional FGM composite thick plates are presented after some validation case studies to show the applicability of PGD in such problems.

Journal ArticleDOI
12 Apr 2021
TL;DR: The ESI Group’s aim is to provide real-time information about the physical properties of the Saarinen Tower and its surroundings to help engineers and scientists better understand the structure and purpose of the building.
Abstract: The need of solving industrial problems using faster and less computationally expensive techniques is becoming a requirement to cope with the present digital transformation of most industries. Recently, data is conquering the domain of engineering with different purposes: (i) defining data-driven models of materials, processes, structures and systems, whose physics-based models, when they exists, remain too inaccurate; (ii) enriching the existing physics-based models within the so-called hybrid paradigm; and (iii) using advanced machine learning and artificial intelligence techniques for scales bridging (upscaling), that is, for creating models that operating at the coarse-grained scale (cheaper in what respect the computational resources) enables integrating the fine-scale richness. The present work addresses the last item, aiming at enhancing standard structural models (defined in 2D shell geometries) for accounting all the fine-scale details (3D with rich through-the-thickness behaviors). For this purpose, two main strategies will be combined: (i) the in-plane-out-of-plane proper generalized decomposition -PGD- serving to provide the fine-scale richness; and (ii) advance machine learning techniques able to learn and extract the regression relating the input parameters with those high-resolution detailed descriptions.

Cites background from "A new hybrid explicit/implicit in-p..."

  • ...[1-5]; (ii) thermal models defined in plates and laminates [6-7]; (iii) flows of Newtonian and non-Newtonian fluids in thin flat and rough gaps [8-12]; (iv) electromagnetism in stratified composites [13]; ....

    [...]


References
More filters

Book
26 Jun 1995
Abstract: 1. An Introduction to the Use of Finite Element Procedures. 2. Vectors, Matrices and Tensors. 3. Some Basic Concepts of Engineering Analysis and an Introduction to the Finite Element Methods. 4. Formulation of the Finite Element Method -- Linear Analysis in Solid and Structural Mechanics. 5. Formulation and Calculation of Isoparametric Finite Element Matrices. 6. Finite Element Nonlinear Analysis in Solid and Structural Mechanics. 7. Finite Element Analysis of Heat Transfer, Field Problems, and Incompressible Fluid Flows. 8. Solution of Equilibrium Equations in State Analysis. 9. Solution of Equilibrium Equations in Dynamic Analysis. 10. Preliminaries to the Solution of Eigenproblems. 11. Solution Methods for Eigenproblems. 12. Implementation of the Finite Element Method. References. Index.

7,875 citations


"A new hybrid explicit/implicit in-p..." refers methods in this paper

  • ...Before introducing the hybrid strategy we consider at time tkþ1 the standard implicit and explicit formulations (two commun time integration schemas among other possibilities [4]), given respectively by...

    [...]


Journal ArticleDOI
TL;DR: This work states thatKinetic theory models involving the Fokker-Planck equation can be accurately discretized using a mesh support using a reduced approximation basis within an adaptive procedure making use of an efficient separation of variables.
Abstract: Kinetic theory models involving the Fokker-Planck equation can be accurately discretized using a mesh support (finite elements, finite differences, finite volumes, spectral techniques, etc.). However, these techniques involve a high number of approximation functions. In the finite element framework, widely used in complex flow simulations, each approximation function is related to a node that defines the associated degree of freedom. When the model involves high dimensional spaces (including physical and conformation spaces and time), standard discretization techniques fail due to an excessive computation time required to perform accurate numerical simulations. One appealing strategy that allows circumventing this limitation is based on the use of reduced approximation basis within an adaptive procedure making use of an efficient separation of variables. (c) 2006 Elsevier B.V. All rights reserved.

511 citations


Journal ArticleDOI
Thomas J. R. Hughes, Wing Kam Liu1Institutions (1)

327 citations


"A new hybrid explicit/implicit in-p..." refers background in this paper

  • ...In [18] a hybrid schema was proposed that considers the domain composed of two parts in which explicit and implicit time integrations apply....

    [...]


Journal ArticleDOI
TL;DR: This work presents a new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids using separated representations and tensor product approximations basis for treating transient models.
Abstract: Kinetic theory models described within the Fokker-Planck formalism involve high-dimensional spaces (including physical and conformation spaces and time). One appealing strategy for treating this kind of problems lies in the use of separated representations and tensor product approximations basis. This technique that was introduced in a former work [A. Ammar, B. Mokdad, E Chinesta, R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, J. Non-Newtonian Fluid Mech. 139 (2006) 153-176] for treating steady state kinetic theory models is extended here for treating transient models. (c) 2007 Elsevier B.V. All rights reserved.

307 citations


"A new hybrid explicit/implicit in-p..." refers methods in this paper

  • ...Even if as also indicated space-time separated discretizations were considered many times in the past [20,2], in the present work time derivatives are discretized using standard schemes....

    [...]


Book
08 Oct 2013
TL;DR: The present text is the first available book describing the Proper Generalized Decomposition (PGD), and provides a very readable and practical introduction that allows the reader to quickly grasp the main features of the method.
Abstract: Many problems in scientific computing are intractable with classical numerical techniques. These fail, for example, in the solution of high-dimensional models due to the exponential increase of the number of degrees of freedom.Recently, the authors of this book and their collaborators have developed a novel technique, called Proper Generalized Decomposition (PGD) that has proven to be a significant step forward. The PGD builds by means of a successive enrichment strategy a numerical approximation of the unknown fields in a separated form. Although first introduced and successfully demonstrated in the context of high-dimensional problems, the PGD allows for a completely new approach for addressing more standard problems in science and engineering. Indeed, many challenging problems can be efficiently cast into a multi-dimensional framework, thus opening entirely new solution strategies in the PGD framework. For instance, the material parameters and boundary conditions appearing in a particular mathematical model can be regarded as extra-coordinates of the problem in addition to the usual coordinates such as space and time. In the PGD framework, this enriched model is solved only once to yield a parametric solution that includes all particular solutions for specific values of the parameters. The PGD has now attracted the attention of a large number of research groups worldwide. The present text is the first available book describing the PGD. It provides a very readable and practical introduction that allows the reader to quickly grasp the main features of the method. Throughout the book, the PGD is applied to problems of increasing complexity, and the methodology is illustrated by means of carefully selected numerical examples. Moreover, the reader has free access to the Matlab software used to generate these examples.

297 citations


"A new hybrid explicit/implicit in-p..." refers background in this paper

  • ...The interested reader can refer to the primer [9] and the numerous references therein....

    [...]


Performance
Metrics
No. of citations received by the Paper in previous years
YearCitations
20213
20192