This paper explores an alternative to shell computation. The proposed strategy uses the Proper Generalized Methods based on a separated representation. The idea is to solve the full 3D solid problem separating the in-plane and the out-of-plane spaces. This allows to represents complex fields in the thickness without the complexity and the computational cost of a solid mesh which is particularly interesting when dealing with multi-layer composite

METEOUI, Sondes

PRULIERE, Etienne

This paper explores an alternative to shell computation. The proposed strategy uses the Proper Generalized Methods based on a separated representation. The idea is to solve the full 3D solid problem separating the in-plane and the out-of-plane spaces. This allows to represents complex ﬁelds in the thickness without the complexity and the computational cost of a solid mesh which is particularly interesting when dealing with multi-layer composite

PRULIÈRE, Etienne

METOUI, Sondes

Oskar Bordeaux

A reduced numerical strategy based on PGD for composite shell structures simulations

SAM : Science Arts et Métiers

Science Arts & Métiers (SAM)is an open access repository that collects the work of Arts et Métiers ParisTechresearchers and makes it freely available over the web where possible.This is an author-deposited version published in: http://sam.ensam.euHandle ID: .http://hdl.handle.net/10985/7879To cite this version :Etienne PRULIERE, Sondes METEOUI - A reduced numerical strategy based on PGD forcomposite shell structures simulations - 2013Any correspondence concerning this service should be sent to the repositoryAdministrator : archiveouverte@ensam.euYIC2013Second ECCOMAS Young Investigators Conference2–6 September 2013, Bordeaux, FranceA reduced numerical strategy based on PGD for composite shell structuressimulationsE. Prulie`re a,∗, S. Metoui a,a Intitut de me´canique et d’inge´nierie de BordeauxArts et Me´tiers ParisTech, Esplanade des Arts et Me´tiers, 33400 Talence, France∗e.pruliere@i2m.u-bordeaux1.frAbstract. This paper explores an alternative to shell computation. The proposed strategy uses the Proper Gen-eralized Methods based on a separated representation. The idea is to solve the full 3D solid problem separatingthe in-plane and the out-of-plane spaces. This allows to represents complex fields in the thickness without the com-plexity and the computational cost of a solid mesh which is particularly interesting when dealing with multi-layercomposite.Keywords: Shell structures; Composites; Proper Generalized Decomposition.1 INTRODUCTIONThe composite structures are more and more used because of the need in structures lightening especially in aeronau-tical applications. The damaging process in composite structures is of great complexity especially under dynamicsolicitations. Organic matrix composites structures are generally composed of three characteristic scales. The mi-croscopic scale is related to the fibres arrangement in the matrix, the mesoscopic scale is related to the plies and themacroscopic scale is related to the structures. Damage can occur at these three scales:• fibres ruptures, fibres/matrix decohesion and matrix cracking at microscopic scale,• delamination at mesoscopic scale,• macroscopic ruptures.Thus, the damaging of composites is a multidimensional phenomenon. An efficient model must be able to considerthe different damaging types and their scales and the interaction between them. Another constrain to a good predic-tion of damages is that a good model must be able to be solved with numerical solver. With the classical numericaltools, it is definitely not conceivable to perform a complete multiscale simulation considering fibres and structures.2 NUMERICAL STRATEGIES2.1 Restriction of classic numerical strategiesThe mainly used numerical method for mechanic simulations is the finite element method (FEM). For example,numerical simulations of composite structures are generally performed using multi-layered shell elements in thecontext of the finite elements method. This strategy has numerous advantages like a low computation time and thecapability to reproduce the comportment of composites in most of cases. The main restriction of this approach isthat it has only a coarse description of strain and stresses variations in the thickness. This approximation is no morevalid when increasing the thickness, near the boundary and loading conditions and when non linear phenomena likedelamination or other damaging processes occurs in the thickness. Consequently other strategies must be developedfor damaging simulations.2 E. Prulie`re et al. | Young Investigators Conference 20132.2 The Proper Generalized DecompositionModel reduction techniques are being developed in computational mechanics. This kind of strategies has proveits efficacy in many applications like thermal problems [3], stochastic simulations, rheology ([1], [2], [4]) andstructural mechanics [5]. This techniques are based on a separated representation of the solution. Methods based onthe proper orthognal decomposition (POD) needs a precomputation to build a reduced approximation basis and tofind the solution. So it is needed to know the solution prior to compute the approximation basis. Others strategieslikes the A priori Hyper Reduction Method, allows building the spacial approximation basis and the solution at thesame time. Another method consist in searching the solution directly on a defined separated form. This method iscalled the Proper Generalized Decomposition (PGD). Details on the method can be found here [?]. A resolutionwith the PGD needs:1. An equation to solve, i.e. a partial differential equation modelling the physical phenomena.2. A separated decomposition of the unknown in order to build the solution.3. An approximation of the solution on each subspace (a Finite Element approximation is often used).4. Some boundary conditions defined on the subspaces (the application of boundary conditions may be a littlecomplicated)3 USING THE PGD ON SHELL STRUCTURES3.1 Mechanical modelThe model used is based on the classical momentum conservation equation:divσ + f = ρΓ (1)where σ denotes the stress tensor, div denotes the tensorial divergence,f is the volume force, ρ is the density and Γis the acceleration. σ is linked to the deformation by the constitutive relation:σ = Hε (2)where ε is the strain tensor and H is the fourth order rigidity tensor.ε is the symmetric gradient of the displacement u = (u, v, w).The weak formulation of the equilibrium equation Eq. (??) without dynamic effect is:∫Ωε(u) : (Hε(u)) =∫Ωu.f +∫∂Ωu.(σ.n) (3)where Ω is the domain taken by the structure.3.2 Separated representationThe idea is to use the PGD method separating a solid shell structure in two spaces: the mid-plane surface andthe thickness. The shell structure occupies a region Ω = S × T where S describes the mid-plane surface andT = [− e2 ; e2 ] is an interval containing all the positions in the thickness defined by a signed distance from themid-plane. e denotes the maximum thickness of the shell.⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩u(X, z) ≈N∑i=1Fui (X)Gui (z)v(X, z) ≈N∑i=1F vi (X)Gvi (z)w(X, z) ≈N∑i=1Fwi (X)Gwi (z)∀X ∈ S (4)X is the position vector and z is the local out-of-plane coordinate. The vector function Fi = (Fui , Fvi , Fwi ) isdefined over the mid-plane surface S and the vector function Gi = (Gui , Gvi , Gwi ) is defined over the thickness ofthe shell structure.E. Prulie`re et al. | Young Investigators Conference 2013 3Figure 1: Example of a triangular shell mesh for the mid plane surface (left) and a 1D mesh for the thickness (right).3.3 Approximation on the subspaceThe vector functions Fi are approximated using shell or a plate finite elements. The shell mesh describe the positionon the mid-plane. The Gi functions are approximated by simple 1D finite elements describing the position on thethickness. It can be noticed that a 3D structured mesh of the shell structure can be derived from the two previousmeshes (depicted in Figure 1).3.4 References formatThe bibliography should be the last section of the paper and use the layout as given in this example and 9pt font.Use an unnumbered heading ‘References’ in the primary heading format. For a book reference use example [?]. Fora journal article use [?]. An article in proceedings should look like [?]. The references should be labeled in the orderin which they appear in the text. Place references in the text using a number or a list of numbers between squarebrackets.4 CONCLUSIONSIn this brief article, a promising numerical strategy has been exposed. This strategy permits a full 3D modeling ofstructures with the computational complexity of a non linear shell modelling. This approach is interesting in orderto simulate composites damaging with a low computational coast.REFERENCES[1] A. Ammar, B. Mokdad, F. Chinesta, R. Keunings. A New Family of Solvers for Some Classes of Multidimensional PartialDifferential Equations Encountered in Kinetic Theory Modeling of Complex Fluids. J. Non-Newtonian Fluid Mech. 139,153-176, 2006.[2] A. Ammar, B. Mokdad, F. Chinesta, R. Keunings. A New Family of Solvers for Some Classes of Multidimensional PartialDifferential Equations Encountered in Kinetic Theory Modeling of Complex Fluids. Part II: Transient Simulation UsingSpace-Time Separated Representation. J. Non-Newtonian Fluid Mech. 144, 98-121, 2007.[3] F. Chinesta, A. Ammar, F. Lemarchand, P. Beauchene, F. Boust, Alleviating mesh constraints: Model reduction, paralleltime integration and high resolution homogenization, Comput. Methods Appl. Mech. Engrg. 197, 400-413, 2008.[4] E. Prulie`re, A. Ammar, N. El Kissi, F. Chinesta. Recirculating flows involving short fiber suspensions: numerical difficultiesand efficient micro-macro solvers. Archives of Computational Methods in Engineering, 16:1, 1-30, 2009.[5] E. Prulire, J. Frec, F. Chinesta, A. Ammar. An efficient reduced simulation of residual stresses in composite formingprocesses. Int. J. of Material Forming, 3:1, 2010.

English

A New Family of Solvers for Some Classes of Multidimensional Partial Differential Equations Encountered in Kinetic Theory Modeling of Complex Fluids.

A New Family of Solvers for Some Classes of Multidimensional Partial Differential Equations Encountered in Kinetic Theory Modeling of Complex Fluids. Part II: Transient Simulation Using Space-Time Separated Representation.

A.Ammar, N.ElKissi, F.Chinesta.Recirculatingﬂowsinvolvingshortﬁbersuspensions: numericaldifﬁculties and efﬁcient micro-macro solvers.

Alleviating mesh constraints: Model reduction, parallel time integration and high resolution homogenization,

An efﬁcient reduced simulation of residual stresses in composite forming processes.

Name not available

Science Arts & Métiers (SAM)is an open access repository that collects the work of Arts et Métiers ParisTechresearchers and makes it freely available over the web where possible.This is an author-deposited version published in: http://sam.ensam.euHandle ID: .http://hdl.handle.net/10985/7879To cite this version :Etienne PRULIERE, Sondes METEOUI - A reduced numerical strategy based on PGD forcomposite shell structures simulations - 2013Any correspondence concerning this service should be sent to the repositoryAdministrator : archiveouverte@ensam.euYIC2013Second ECCOMAS Young Investigators Conference2–6 September 2013, Bordeaux, FranceA reduced numerical strategy based on PGD for composite shell structuressimulationsE. Prulie`re a,∗, S. Metoui a,a Intitut de me´canique et d’inge´nierie de BordeauxArts et Me´tiers ParisTech, Esplanade des Arts et Me´tiers, 33400 Talence, France∗e.pruliere@i2m.u-bordeaux1.frAbstract. This paper explores an alternative to shell computation. The proposed strategy uses the Proper Gen-eralized Methods based on a separated representation. The idea is to solve the full 3D solid problem separatingthe in-plane and the out-of-plane spaces. This allows to represents complex ﬁelds in the thickness without the com-plexity and the computational cost of a solid mesh which is particularly interesting when dealing with multi-layercomposite.Keywords: Shell structures; Composites; Proper Generalized Decomposition.1 INTRODUCTIONThe composite structures are more and more used because of the need in structures lightening especially in aeronau-tical applications. The damaging process in composite structures is of great complexity especially under dynamicsolicitations. Organic matrix composites structures are generally composed of three characteristic scales. The mi-croscopic scale is related to the ﬁbres arrangement in the matrix, the mesoscopic scale is related to the plies and themacroscopic scale is related to the structures. Damage can occur at these three scales:• ﬁbres ruptures, ﬁbres/matrix decohesion and matrix cracking at microscopic scale,• delamination at mesoscopic scale,• macroscopic ruptures.Thus, the damaging of composites is a multidimensional phenomenon. An efﬁcient model must be able to considerthe different damaging types and their scales and the interaction between them. Another constrain to a good predic-tion of damages is that a good model must be able to be solved with numerical solver. With the classical numericaltools, it is deﬁnitely not conceivable to perform a complete multiscale simulation considering ﬁbres and structures.2 NUMERICAL STRATEGIES2.1 Restriction of classic numerical strategiesThe mainly used numerical method for mechanic simulations is the ﬁnite element method (FEM). For example,numerical simulations of composite structures are generally performed using multi-layered shell elements in thecontext of the ﬁnite elements method. This strategy has numerous advantages like a low computation time and thecapability to reproduce the comportment of composites in most of cases. The main restriction of this approach isthat it has only a coarse description of strain and stresses variations in the thickness. This approximation is no morevalid when increasing the thickness, near the boundary and loading conditions and when non linear phenomena likedelamination or other damaging processes occurs in the thickness. Consequently other strategies must be developedfor damaging simulations.2 E. Prulie`re et al. | Young Investigators Conference 20132.2 The Proper Generalized DecompositionModel reduction techniques are being developed in computational mechanics. This kind of strategies has proveits efﬁcacy in many applications like thermal problems [3], stochastic simulations, rheology ([1], [2], [4]) andstructural mechanics [5]. This techniques are based on a separated representation of the solution. Methods based onthe proper orthognal decomposition (POD) needs a precomputation to build a reduced approximation basis and toﬁnd the solution. So it is needed to know the solution prior to compute the approximation basis. Others strategieslikes the A priori Hyper Reduction Method, allows building the spacial approximation basis and the solution at thesame time. Another method consist in searching the solution directly on a deﬁned separated form. This method iscalled the Proper Generalized Decomposition (PGD). Details on the method can be found here [?]. A resolutionwith the PGD needs:1. An equation to solve, i.e. a partial differential equation modelling the physical phenomena.2. A separated decomposition of the unknown in order to build the solution.3. An approximation of the solution on each subspace (a Finite Element approximation is often used).4. Some boundary conditions deﬁned on the subspaces (the application of boundary conditions may be a littlecomplicated)3 USING THE PGD ON SHELL STRUCTURES3.1 Mechanical modelThe model used is based on the classical momentum conservation equation:divσ + f = ρΓ (1)where σ denotes the stress tensor, div denotes the tensorial divergence,f is the volume force, ρ is the density and Γis the acceleration. σ is linked to the deformation by the constitutive relation:σ = Hε (2)where ε is the strain tensor and H is the fourth order rigidity tensor.ε is the symmetric gradient of the displacement u = (u, v, w).The weak formulation of the equilibrium equation Eq. (??) without dynamic effect is:∫Ωε(u) : (Hε(u)) =∫Ωu.f +∫∂Ωu.(σ.n) (3)where Ω is the domain taken by the structure.3.2 Separated representationThe idea is to use the PGD method separating a solid shell structure in two spaces: the mid-plane surface andthe thickness. The shell structure occupies a region Ω = S × T where S describes the mid-plane surface andT = [− e2 ; e2 ] is an interval containing all the positions in the thickness deﬁned by a signed distance from themid-plane. e denotes the maximum thickness of the shell.⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩u(X, z) ≈N∑i=1Fui (X)Gui (z)v(X, z) ≈N∑i=1F vi (X)Gvi (z)w(X, z) ≈N∑i=1Fwi (X)Gwi (z)∀X ∈ S (4)X is the position vector and z is the local out-of-plane coordinate. The vector function Fi = (Fui , Fvi , Fwi ) isdeﬁned over the mid-plane surface S and the vector function Gi = (Gui , Gvi , Gwi ) is deﬁned over the thickness ofthe shell structure.E. Prulie`re et al. | Young Investigators Conference 2013 3Figure 1: Example of a triangular shell mesh for the mid plane surface (left) and a 1D mesh for the thickness (right).3.3 Approximation on the subspaceThe vector functions Fi are approximated using shell or a plate ﬁnite elements. The shell mesh describe the positionon the mid-plane. The Gi functions are approximated by simple 1D ﬁnite elements describing the position on thethickness. It can be noticed that a 3D structured mesh of the shell structure can be derived from the two previousmeshes (depicted in Figure 1).3.4 References formatThe bibliography should be the last section of the paper and use the layout as given in this example and 9pt font.Use an unnumbered heading ‘References’ in the primary heading format. For a book reference use example [?]. Fora journal article use [?]. An article in proceedings should look like [?]. The references should be labeled in the orderin which they appear in the text. Place references in the text using a number or a list of numbers between squarebrackets.4 CONCLUSIONSIn this brief article, a promising numerical strategy has been exposed. This strategy permits a full 3D modeling ofstructures with the computational complexity of a non linear shell modelling. This approach is interesting in orderto simulate composites damaging with a low computational coast.REFERENCES[1] A. Ammar, B. Mokdad, F. Chinesta, R. Keunings. A New Family of Solvers for Some Classes of Multidimensional PartialDifferential Equations Encountered in Kinetic Theory Modeling of Complex Fluids. J. Non-Newtonian Fluid Mech. 139,153-176, 2006.[2] A. Ammar, B. Mokdad, F. Chinesta, R. Keunings. A New Family of Solvers for Some Classes of Multidimensional PartialDifferential Equations Encountered in Kinetic Theory Modeling of Complex Fluids. Part II: Transient Simulation UsingSpace-Time Separated Representation. J. Non-Newtonian Fluid Mech. 144, 98-121, 2007.[3] F. Chinesta, A. Ammar, F. Lemarchand, P. Beauchene, F. Boust, Alleviating mesh constraints: Model reduction, paralleltime integration and high resolution homogenization, Comput. Methods Appl. Mech. Engrg. 197, 400-413, 2008.[4] E. Prulie`re, A. Ammar, N. El Kissi, F. Chinesta. Recirculating ﬂows involving short ﬁber suspensions: numerical difﬁcultiesand efﬁcient micro-macro solvers. Archives of Computational Methods in Engineering, 16:1, 1-30, 2009.[5] E. Prulire, J. Frec, F. Chinesta, A. Ammar. An efﬁcient reduced simulation of residual stresses in composite formingprocesses. Int. J. of Material Forming, 3:1, 2010.

SAM;  the Arts et Métiers ParisTech open access repository

Science Arts & Métiers (SAM)
is an open access repository that collects the work of Arts et Métiers ParisTech
researchers and makes it freely available over the web where possible.
This is an author-deposited version published in: https://sam.ensam.eu
Handle ID: .http://hdl.handle.net/10985/7879
To cite this version :
Etienne PRULIERE, Sondes METOUI - A reduced numerical strategy based on PGD for
composite shell structures simulations - 2013
Any correspondence concerning this service should be sent to the repository
Administrator : archiveouverte@ensam.eu
YIC2013
Second ECCOMAS Young Investigators Conference
2–6 September 2013, Bordeaux, France
A reduced numerical strategy based on PGD for composite shell structures
simulations
E. Prulie`re a,∗, S. Metoui a,
a Intitut de me´canique et d’inge´nierie de Bordeaux
Arts et Me´tiers ParisTech, Esplanade des Arts et Me´tiers, 33400 Talence, France
∗e.pruliere@i2m.u-bordeaux1.fr
Abstract. This paper explores an alternative to shell computation. The proposed strategy uses the Proper Gen-
eralized Methods based on a separated representation. The idea is to solve the full 3D solid problem separating
the in-plane and the out-of-plane spaces. This allows to represents complex ﬁelds in the thickness without the com-
plexity and the computational cost of a solid mesh which is particularly interesting when dealing with multi-layer
composite.
Keywords: Shell structures; Composites; Proper Generalized Decomposition.
1 INTRODUCTION
The composite structures are more and more used because of the need in structures lightening especially in aeronau-
tical applications. The damaging process in composite structures is of great complexity especially under dynamic
solicitations. Organic matrix composites structures are generally composed of three characteristic scales. The mi-
croscopic scale is related to the ﬁbres arrangement in the matrix, the mesoscopic scale is related to the plies and the
macroscopic scale is related to the structures. Damage can occur at these three scales:
• ﬁbres ruptures, ﬁbres/matrix decohesion and matrix cracking at microscopic scale,
• delamination at mesoscopic scale,
• macroscopic ruptures.
Thus, the damaging of composites is a multidimensional phenomenon. An efﬁcient model must be able to consider
the different damaging types and their scales and the interaction between them. Another constrain to a good predic-
tion of damages is that a good model must be able to be solved with numerical solver. With the classical numerical
tools, it is deﬁnitely not conceivable to perform a complete multiscale simulation considering ﬁbres and structures.
2 NUMERICAL STRATEGIES
2.1 Restriction of classic numerical strategies
The mainly used numerical method for mechanic simulations is the ﬁnite element method (FEM). For example,
numerical simulations of composite structures are generally performed using multi-layered shell elements in the
context of the ﬁnite elements method. This strategy has numerous advantages like a low computation time and the
capability to reproduce the comportment of composites in most of cases. The main restriction of this approach is
that it has only a coarse description of strain and stresses variations in the thickness. This approximation is no more
valid when increasing the thickness, near the boundary and loading conditions and when non linear phenomena like
delamination or other damaging processes occurs in the thickness. Consequently other strategies must be developed
for damaging simulations.
2 E. Prulie`re et al. | Young Investigators Conference 2013
2.2 The Proper Generalized Decomposition
Model reduction techniques are being developed in computational mechanics. This kind of strategies has prove
its efﬁcacy in many applications like thermal problems [3], stochastic simulations, rheology ([1], [2], [4]) and
structural mechanics [5]. This techniques are based on a separated representation of the solution. Methods based on
the proper orthognal decomposition (POD) needs a precomputation to build a reduced approximation basis and to
ﬁnd the solution. So it is needed to know the solution prior to compute the approximation basis. Others strategies
likes the A priori Hyper Reduction Method, allows building the spacial approximation basis and the solution at the
same time. Another method consist in searching the solution directly on a deﬁned separated form. This method is
called the Proper Generalized Decomposition (PGD). Details on the method can be found here [?]. A resolution
with the PGD needs:
1. An equation to solve, i.e. a partial differential equation modelling the physical phenomena.
2. A separated decomposition of the unknown in order to build the solution.
3. An approximation of the solution on each subspace (a Finite Element approximation is often used).
4. Some boundary conditions deﬁned on the subspaces (the application of boundary conditions may be a little
complicated)
3 USING THE PGD ON SHELL STRUCTURES
3.1 Mechanical model
The model used is based on the classical momentum conservation equation:
divσ + f = ρΓ (1)
where σ denotes the stress tensor, div denotes the tensorial divergence,f is the volume force, ρ is the density and Γ
is the acceleration. σ is linked to the deformation by the constitutive relation:
σ = Hε (2)
where ε is the strain tensor and H is the fourth order rigidity tensor.
ε is the symmetric gradient of the displacement u = (u, v, w).
The weak formulation of the equilibrium equation Eq. (??) without dynamic effect is:
∫
Ω
ε(u) : (Hε(u)) =
∫
Ω
u.f +
∫
∂Ω
u.(σ.n) (3)
where Ω is the domain taken by the structure.
3.2 Separated representation
The idea is to use the PGD method separating a solid shell structure in two spaces: the mid-plane surface and
the thickness. The shell structure occupies a region Ω = S × T where S describes the mid-plane surface and
T = [− e2 ; e2 ] is an interval containing all the positions in the thickness deﬁned by a signed distance from the
mid-plane. e denotes the maximum thickness of the shell.
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
u(X, z) ≈
N∑
i=1
Fui (X)G
u
i (z)
v(X, z) ≈
N∑
i=1
F vi (X)G
v
i (z)
w(X, z) ≈
N∑
i=1
Fwi (X)G
w
i (z)
∀X ∈ S (4)
X is the position vector and z is the local out-of-plane coordinate. The vector function Fi = (Fui , F
v
i , F
w
i ) is
deﬁned over the mid-plane surface S and the vector function Gi = (Gui , Gvi , Gwi ) is deﬁned over the thickness of
the shell structure.
E. Prulie`re et al. | Young Investigators Conference 2013 3
Figure 1: Example of a triangular shell mesh for the mid plane surface (left) and a 1D mesh for the thickness (right).
3.3 Approximation on the subspace
The vector functions Fi are approximated using shell or a plate ﬁnite elements. The shell mesh describe the position
on the mid-plane. The Gi functions are approximated by simple 1D ﬁnite elements describing the position on the
thickness. It can be noticed that a 3D structured mesh of the shell structure can be derived from the two previous
meshes (depicted in Figure 1).
3.4 References format
The bibliography should be the last section of the paper and use the layout as given in this example and 9pt font.
Use an unnumbered heading ‘References’ in the primary heading format. For a book reference use example [?]. For
a journal article use [?]. An article in proceedings should look like [?]. The references should be labeled in the order
in which they appear in the text. Place references in the text using a number or a list of numbers between square
brackets.
4 CONCLUSIONS
In this brief article, a promising numerical strategy has been exposed. This strategy permits a full 3D modeling of
structures with the computational complexity of a non linear shell modelling. This approach is interesting in order
to simulate composites damaging with a low computational coast.
REFERENCES
[1] A. Ammar, B. Mokdad, F. 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,
153-176, 2006.
[2] A. Ammar, B. Mokdad, F. Chinesta, R. Keunings. A New Family of Solvers for Some Classes of Multidimensional Partial
Differential Equations Encountered in Kinetic Theory Modeling of Complex Fluids. Part II: Transient Simulation Using
Space-Time Separated Representation. J. Non-Newtonian Fluid Mech. 144, 98-121, 2007.
[3] F. Chinesta, A. Ammar, F. Lemarchand, P. Beauchene, F. Boust, Alleviating mesh constraints: Model reduction, parallel
time integration and high resolution homogenization, Comput. Methods Appl. Mech. Engrg. 197, 400-413, 2008.
[4] E. Prulie`re, A. Ammar, N. El Kissi, F. Chinesta. Recirculating ﬂows involving short ﬁber suspensions: numerical difﬁculties
and efﬁcient micro-macro solvers. Archives of Computational Methods in Engineering, 16:1, 1-30, 2009.
[5] E. Prulire, J. Frec, F. Chinesta, A. Ammar. An efﬁcient reduced simulation of residual stresses in composite forming
processes. Int. J. of Material Forming, 3:1, 2010.


A reduced numerical strategy based on PGD for composite shell structures simulations

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