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Combined Topology and Stacking Sequence Optimization of Composite Laminated Structures for Structural Performance Measure

G.P. Rodrigues, J. Folgado, J.M. Guedes IDMEC, IST, Lisbon ENGOPT2014

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Summary

• Composite materials
• Discrete Material Optimization

(DMO)

• Maximize stiffness
• Sensitivity analisys
• Finite Element Analisys – Abaqus
• Feasible Arc Interior Point Algorithm

(FAIPA)

• Results and conclusions

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Discrete Material Optimization*

• Transforms the discrete optimization problem into a

continuos one

• Constitutive matrix is composed by a weighted sum of

the matrices from the candidate materials

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• Weights are function of material variables
• Variables and weights tend to limit values of 0 or 1
• DMO scheme 4

*J.Stegmann & E.Lund (2005) Discrete Material Optimization of General Composite Shell Structures

Candidate Materials

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• Composite material

– Orthotropic material – Different fiber orientations – 0º, 90º, 45º and -45º

• Foam material

– Isotropic material – 250x less stiff – 20x less dense

Design regions and convergence

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• Structures divided in areas “j” and laminas “k”
• Laminas and areas corresponde to design regions “r”
• Obtain defined material in each design region
• Material weights to limit values 0 and 1
• Convergence assumed with weights of 0.95

Optimization functions

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Subjected to:

• Box constraints – project values

go between 0 and 1

• Equality constraints – force

weights with limit values (0 and 1)

• Inequality constraints – minimum

number of foam regions

Sensitivity Analisys

• Derivative of the objective function → Adjoint Method

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• Derivative of the equality and inequality constraints

Feasible Arc Interior Point Algorithm (FAIPA)

• Programmed in Matlab
• Requires initial feasible point
• Considers KKT conditions for optimality
• Line search: Armijo, Wolfe and Goldstein
• Uses “Feasible direction” or “Feasible descent arc”
• Permits Newton, Quasi-Newton and First Order

Methods

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Abaqus – Finite Element Model

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Advantages:

– General FEA code and permits to study complex/generic structures – Numerous shell elements available (e.g. S3, S4, S4R and SAX1)

Disavantages:

– Computacionally heavy – Works as a black box – Difficulties to extract data

S4 Shell element:

– 4 node shell element based on FSDT – Without reduced integration (with capabilities to prevent locking ) – Allows the user to define layers – Recomended for thick shells and composite thin shells – May not perform very well for sandwich laminates

Penalty function

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• FAIPA uses Feasible solution method
• Difficulties in convergence due to unfeasible starting

points

• Introduce penalized functions in FAIPA

Program description

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Develop sructural model Define

• ptimization

parameters Pre-Optimizer Optimization (FAIPA) Results

Pre-Optimizer:

– Call Matlab – Initialize parameters – Read Abaqus input file (write assembly matrix) – Call Abaqus (obtain stiffness matrices)

Optimization (FAIPA):

– Call Abaqus and obtain solution – Read output – Calculate derivatives – Increment penalty exponent and coeficients – Iterate until convergence

Results:

– Writes in Matlab, Excel and text files – Design points, objective values and material weights

Results (Laminate)

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• Square plate with 10 laminas
• Subjected to uniform pressure (P) and variable

traction load (T)

• Include 2 foam layers

Results (Combined Topology)

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• Foam material can be adjusted to become “weaker”

and perform topology optimization

• Square plate with 400 areas
• Case studies:

– Concentrated load in fixed plate (a) – Central pressure in simply supported plate (b)

Results (Combined Topology (a))

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100 foam areas 200 foam areas 300 foam areas

• DMO allows us to perform topology optimization
• The more foam we introduce the more difficult it is to

achieve convergence

Results (Combined Topology (b))

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• Shows the areas to be reinforced with composite

materials

Results (different coordinate systems)

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• 2 plate areas with

different local coordinate systems

• Each area with 5 layers
• Fixed in one edge
• Case studies:

– Concentrated load – Traction and bending – Shear edge load

• Include foam

Results (different coordinate systems (a))

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Results (different coordinate systems (b)

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Results (different coordinate systems)

Discussion and Conclusion

• General good results and high convergence
• DMO creates many local minimums and design variables
• A proper parameters choice has to be performed (the

designer experience is relevant)

• Penalty function with FAIPA performs well
• Code developed is limited to S4 elements but easilly extended

to other

• For higher amounts of foam convergence is more difficult
• Equality constraints helps to overcome some material

mixtures

• Extend to multi-load optimization
• Adapt code to various kinds of other elements

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Acknowledgements

This work is supported by the Project FCT PT DC/EME-PME /120630/2010

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