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Support for Decision Making in Design of Composite Laminated Structures. Part 2: Reduced Parametric Model-Based Optimization

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Abstract

The design process of laminated composites faces two challenges: the engineer designs the product and its morphology, but also, simultaneously, the material. The number of design solutions can be huge since the solution space is very large. Standard CAE systems (CAD, Finite Element Simulation) do not offer to the designer an approach to explore these solution spaces efficiently and interactively. This paper provides a possible procedure for engineers having a laminated composite product to create: it presents an approach that allows combining to usual morphological design parameters, specific variables that are typically the domain of composite experts, and manufacturing experts. Using an optimization approach based on an evolutionary algorithm coupled to a reduced order analysis, a decision support solution is detailed. The numerical approach allows the engineer to explore interactively design spaces. Our approach is consisting in processing a Knowledge Model having a reduced and separated form [9]. We present a decision support method that allow designers to have, both, a multiscale and a multiphysical view on the laminated structures that they are creating. Two design problems are presented to illustrate the relevance of the approach when designing composite structures: one under a static load and the having a dynamic behavior.

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Abbreviations

E :

Young’s modulus (MPa)

E f :

Fiber Young’s modulus (MPa)

E m :

Matrix Young’s modulus (MPa)

E l :

Young’s modulus of the ply in the direction of the fibers (MPa)

E t :

Young’s modulus of the ply in a direction transversal to the fiber direction (MPa)

F y ,F z :

External forces (N)

F(ω) :

Force as a function of frequency (N)

B :

Body force

G :

Shear modulus (MPa)

v :

Poisson’s ratio

l :

Length (mm)

h :

Height (mm)

w :

Width (mm)

u :

Displacement in direction x

v :

Displacement in direction y

w :

Displacement in direction z

ε :

Strain tensor

σ :

Stress Tensor (Pa)

σ ij :

Stress tensor element (Pa)

F 0 :

Objective function

ς, ξ, ψ :

(Weights)

\( {\mathbf{\mathcal{L}}}_{\boldsymbol{max}} \) :

Maximum deformation to the direction y

\( {\boldsymbol{U}}_{\boldsymbol{x},\boldsymbol{y},\boldsymbol{z},{\boldsymbol{p}}_{\mathbf{1}},{\boldsymbol{p}}_{\mathbf{2}},\bullet \bullet \bullet, {\boldsymbol{p}}_{\boldsymbol{d}}} \) :

Approximation of displacement field (mm)

U(x, y, z, p 1, p 1, ⋯, p d) :

Displacement field as a function of given parameters (mm)

C :

Tensor of material properties in local coordinates

n :

Number of enrichment modes in PGD sense

\( \overline{\boldsymbol{C}} \) :

Tensor of material properties in global coordinates

\( \overline{\boldsymbol{C}}\left({\boldsymbol{p}}_{\mathbf{1}}\right),\overline{\boldsymbol{C}}\left({\boldsymbol{p}}_{\mathbf{2}}\right),\overline{\boldsymbol{C}}\left({\boldsymbol{p}}_{\mathbf{3}}\right),\overline{\boldsymbol{C}}\left({\boldsymbol{p}}_{\mathbf{4}}\right) \) :

Tensor of material properties at plies 1, 2, 3, 4 in global coordinates

C int :

Tensor of material properties at the interfaces

D :

Transformation matrix

ρ :

Density (kg/m3)

ρ f :

Fiber density

ρ m :

Resin density

Ω :

Geometric domain

θ i :

Fiber orientation of ply i (degrees)

V f :

Fiber volume fraction (%)

G 0 :

Short term shear modulus (GPa)

G ∞ :

Long-term shear modulus (GPa)

α :

Fractional derivative order

τ :

Decay time (s)

X, Y, Z, P1, P2, P3, P4, P5, P6, P7 :

PGD functions

x,y,z,p 1 ,p 2 ,p 3 ,p 4 ,p 5 ,p 6 ,p 7 :

PGD domains

T max :

Maximum twist

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Acknowledgements

The research was supported by the Colciencias (Colombia) and the Universidad Pontificia Bolivariana (Bucaramanga, Colombia).

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Authors and Affiliations

  1. Université de Bordeaux, I2M CNRS UMR 5295, ENSAM, Esplanade des Arts et Métiers, 33400, Talence, France

    Gilberto Fontecha Dulcey, Xavier Fischer & Pierre Joyot

  2. Université de Bordeaux, École Supérieure des Technologies Industrielles Avancées, ESTIA, technopole izarbel, Techn, 64210, Bidart, Aquitaine, France

    Gilberto Fontecha Dulcey, Xavier Fischer & Pierre Joyot

  3. Universidad Pontificia Bolivariana, Km 7, Autopista Piedecuesta, Bucaramanga, Colombia

    Gilberto Fontecha Dulcey

  4. Clemson University, Mechanical engineering, Fluor Daniel Building, Clemson, S.C, 29634, USA

    Georges Fadel

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  1. Gilberto Fontecha Dulcey
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  4. Georges Fadel
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Correspondence to Xavier Fischer.

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Fontecha Dulcey, G., Fischer, X., Joyot, P. et al. Support for Decision Making in Design of Composite Laminated Structures. Part 2: Reduced Parametric Model-Based Optimization. Appl Compos Mater 26, 663–681 (2019). https://doi.org/10.1007/s10443-018-9742-9

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