Abstract
Designers and engineers are constantly seeking for more efficient structures that can achieve a high performance with minimum weight. Since its inception, topology optimization has been evolving and new features and methods have been proposed to support the development of such lightweight and high-performance structures. In recent years, the application of multiple-material topology optimization in the conceptual design as well as the consideration of composite materials in the topology optimization process has increased. Composite materials usually offer a superior stiffness-to-weight ratio compared to isotropic materials, but it also requires engineers to define the lamination layup. The lamination stacking sequence can also be optimized through different methods, nevertheless the combination of stacking sequence and topology optimization is still limited in the literature. Therefore, the present paper presents a method to simultaneously optimize the material placement, selection and stacking sequence of composite plates in a multi-material topology optimization framework. The methodology considers that the macro-mechanical composite material response is governed by the orientation of the layers and the composite material properties. As a result, the optimum material placement and selection depend on the material layup that is defined along with the optimization process. A series of example problems are presented and discussed to demonstrate the proposed algorithm capabilities.
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This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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Appendix: Pseudocode
Appendix: Pseudocode
! MMTO-SS method.
PROGRAM MAIN
! This function imports all user parameters.
CALL SUBROUTINE IMPORT_USER_PARAMETERS
! Initialize design variables and optimization parameters.
CALL SUBROUTINE INITIALIZE
! Main optimization loop.
DO I = 1, MAXITER.
! Update material properties.
! uses SIMP equation to define the interpolated material tensors.
CALL SUBROUTINE UPDATE_MATERIAL_PROPERTIES
! Run FEA.
CALL SUBROUTINE RUN_FEA
! Compute Objective Function and its Sensitivities.
CALL SUBROUTINE OBJECTIVE_FUNCTION_SENSITIVITIES
! Checkboard filter.
CALL SUBROUTINE CHECKBOARD_FILTER
! Compute Constraint and Sensitivities.
CALL SUBROUTINE CONSTRAINT_FUNCTION_SENSITIVITIES
! MMA design variable update.
CALL SUBROUTINE MMA_DESIGN_VARIABLE_UPDATE
! Write results to ASCII files for users.
CALL SUBROUTINE WRITE_RESULTS
! Set updated design variables from MMA.
CALL SUBROUTINE SET_DESIGN_VARIABLES
! Check for convergence.
CALL SUBROUTINE CONVERGENCE_CHECK
! Exit if convergence is met.
IF(CONVERGENCE_TOLERANCE < = USERDEFINEDVALUE) EXIT.
! Finish Optimization CPU time.
CALL CPU_TIME (FINISH)
ENDDO
END PROGRAM MAIN
! MMTO-SS method – Ply angle retrieval process.
PROGRAM PLY_ANGLE_RETRIEVAL
! This function imports all user parameters – number of layers, optimum set of LPs.
CALL SUBROUTINE IMPORT_USER_PARAMETERS
! This subroutine computes the LPs for each ply orientation and defines its distance from the optimum set of LPs.
CALL SUBROUTINE DATABASE
! This subroutine sorts the LPs candidates.
CALL SUBROUTINE SORT_LPs.
! Write results to ASCII files for users.
CALL SUBROUTINE WRITE_RESULTS
END PROGRAM PLY_ANGLE_RETRIEVAL
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Bohrer, R.Z.G., Kim, I.Y. Multi material topology and stacking sequence optimization of composite laminated plates. Struct Multidisc Optim 65, 274 (2022). https://doi.org/10.1007/s00158-022-03363-1
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DOI: https://doi.org/10.1007/s00158-022-03363-1