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Computational multiscale
Computational multiscale refers to the use of computational capacities together with mathematical models, allowing to account for the structural details of the microscale and coupling them to the macroscale. Computational multiscale approaches are either hierarchical or concurrent. In the hierarchical multiscale the properties are brought from the micro to the macroscale in one direction by Representative Volume Elements (RVE). The concurrent approach is about reevaluating the effects of the microscale at designated conditions; this is the idea behind the two-level Finite Elements (FE2).
Andreassen and Andreassen [34], Sun et al. [35] and De Souza Neto [36] evaluated the theory of elasticity with RVE. Mosby and Matous [37] perform computational homogenization from the microscale using RVE on a high-performance computation (HPC) to simulate mode I delamination. The use RVE to analyze flaws and reinforcements on laminated composites applying the FEM has been adopted by Kushnir and Rabinovitch [38], Puel and Aubry [39], Willot and Jeulin [40], Hu et al. [41], Savvas and Papadopoulos [42], Sakata et al. [43].
The FE2 was presented by Feyel [44] to model the viscoelastic behavior of composite materials coupling microscale representations to macroscopic levels. Unger [45] uses an FE2 on an HPC system to solve an improved procedure to reduce computational time.
The multiscale systems have been largely studied. The book edited by Soutis and Beaumont [46], exposes a good overview of the problems encountered by those interested in the design of composite structures. Notably, most of the authors are regarding the problematic of damage prediction inside the laminate structures. For example:
– Galiotis and Paipetis [47] propose that the damage may occur at the level of interfaces between plies, developing specific criteria to locate the risk of delamination.
– McCartney [48] proposes that the damage may start either with the cracking of the fibers or inner in the matrix, presenting a micro-mechanical model in order to locate the origin of the damage Some other authors consider that damage in laminated structures may be uncertain (Bogdanor et al. [49]), even when it frequently is the consequence of a cyclic behavior (Hosseini et al. [50]), the damage leads to some significant problems of material integrity occurring during the material design process (Beaumont [51]). Consequently, the problem of damage may be modeled better by a micromechanical approach (Wang [52] and Ivancevic [53]).
Representation of laminated structures
From the identification of tasks, specifications, and constraints, to the production of the desired result; the design of a laminated composite follows a standard process similar to the specified by Pahl et al. [54]. This process is illustrated in Figure 1.3 and explained in detail in section 1.8.1 Figure 1.3 also shows the support of simulation capabilities in order to perform many of the computations required by the designer, given the extensive amount of parameters conforming the constitutive laws. During the dimensioning phase, mechanical simulations of composites are a crucial tool, but because of the significant complexity, the reliability of the simulations is still under high uncertainties impacting the costs of experimentation to validate results. For this reason, many researchers have focused their interest in contributing to the improvement of simulation capacities.
Multiphysical point of view in composite structures
A multiphysical approach is considered when the structure exhibits different physical properties, for example, electrical, thermal or mechanical, (Yuan et al. [55]). Within the scope of this thesis, the multiphysical approach is mechanical and goes through the thickness direction in a laminated composite. In this sense, a multiphysical research trend is devoted to understanding the behavior of the interfaces and the effects on the stress transfer (Geers et al. [56]). These effects are interlaminar normal stresses tending to separate the plies, and interlaminar shear stresses tending to slide one ply over the adjacent ones, either because of weak or non-adhesion, both in static and dynamic applications.
Separation and slippage have been studied mainly in laminated structures, for example by Cheng et al. [6] using a spring-layer model, Ramos and Pesce [14] assuming frictionless interfaces in a riser tube, Chattopadhyay et al. [10] introducing fictitious linear springs, Lei et al. [11] experimentally evaluating interface interaction models using micro-Raman spectroscopy, Budiman et al. [13] proposing a fragmentation model or Liu et al. [12] developing a nonlinear cohesive/frictional coupled model. They all use the FEM with contact elements to guarantee continuity. This approach achieves good accuracy, but the consequence is the high computational cost and computing time, this is the reason why these works are concentrated on localized effects.
Among localized effects, delamination is keeping the interest of many research works. Delamination is known to appear in the normal direction to the interface surfaces (mode I) and the in-plane direction to the interfaces (mode II and mode III). The simulation of delamination modes is achieved better by using the Cohesive Zone Model (CZM) because it uses a cohesive zone before the crack tip to avoid the singularity caused by its sharp shape (Figure 1.5).
The PGD method for composite materials
In a lamination, the PGD might also be seen as a multiscale method (Ammar et al., [8], Halabi et al. [131], and Metoui et al. [132]). The method finds a solution first at low scales, considering the details through the thickness direction to iterate basis functions, and then coupling the results at the macro scale along the in-plane direction. Table 1-5 Synthetizes references using the PGD method in laminated composites.
Support to decision making in composite design
Along with the design process, engineers make use of available knowledge to take decisions for the solution to technical problems. The decision-making process also considers requirements and constraints to obtain optimized solutions. Further, the engineer may also use computer support to model and collect information, this is known as Decision Making Support Systems (DMSS) (Pahl et al. [54] Gutierrez et al. [144], Fischer et al. [145]). This concept is represented in Figure 1.10. A DMSS embeds rational models, relevant information or features to assist the engineering decision making, which otherwise would be numerous or impossible for human processing (Bouyssou [146]). By doing so, DMSSs are developed for specific design phases of a composite. For example, in predesign, Hambali et al. [147] developed a DMSS of material selection of composite bumper beams by a hierarchy process; similarly Corona et al. [148] analyse the impact of natural fibers on the life cycle of a composite and Calado et al. [149] worked on a DMSS to select composite materials during the early phases of aircraft structure design. Within the production of composite solutions, Srinivasan et al. [150] use a DMSS to understand composite manufacturing interactions, Sanz-Corretge [151] proposes a decision tree algorithm with many possible laminate combinations and Coronado et al. [152] use a DMSS to evaluate the impact over the supply chain of end products. The attempt to develop a DMSS including all design phases is commonly known as a multiplatform that integrates software modeling the micromechanics and selection of components, CAD, FEM and simulation of enterprise resource planning. Instead of such multiplatform integration, Gascons et al. [153] propose a DMSS integrating all design phases based on a single variable reduction with better efficiency. Moreover, in this thesis (Chapter 5) we use a parametric knowledge model to explore a solution space in the pre-design phase to directly obtain an optimized solution ready to be sent to production.
Table of contents :
CHAPTER 1 MODELING AND DESIGN OF COMPOSITE STRUCTURES: STATE OF THE ART
1.1 INTRODUCTION
1.2 LAMINATED STRUCTURES
1.2.1 Definition
1.2.2 Major parameters
1.3 MULTISCALE APPROACH
1.3.1 Analytical multiscale
1.3.2 Computational multiscale
1.4 REPRESENTATION OF LAMINATED STRUCTURES
1.4.1 Relevance of simulation
1.4.2 Generalities
1.5 MULTIPHYSICAL APPROACH
1.5.1 Multiphysical point of view in composite structures
1.5.2 Viscoelasticity in composite structures
1.6 SIMULATION OF COMPOSITE STRUCTURES, ADVANCES, AND LIMITATIONS
1.6.1 Finite element based simulation
1.6.2 Simulation for multiscale and multiphysical behavior
1.7 MODEL REDUCTION
1.7.1 Methods of reduction
1.7.2 The PGD
1.7.3 The PGD method for composite materials
1.8 DESIGN OF COMPOSITE STRUCTURES
1.8.1 The design process
1.8.2 Design from simulation
PRBM for the interactive optimization of laminated composite structures G. FONTECHA – 2018
1.8.3 Support to decision making in composite design
1.8.4 Optimization leading to the design of laminated composites
1.8.5 Solutions to support the design of composites
1.9 CONCLUSION
CHAPTER 2 REPRESENTATION, BEHAVIOR, AND DESIGN OF COMPOSITE STRUCTURES
2.1 INTRODUCTION
2.2 MULTISCALE MODELING OF COMPOSITE STRUCTURES
2.2.1 3D behavior
2.2.2 Simulation with FEM considering the interfaces between plies.
2.2.3 Behavior and spatial separation
2.2.4 Our multiscale approach
2.3 MULTIPHYSICAL CONSIDERATION OF COMPOSITE STRUCTURES
2.3.1 Justification of creeping in a composite structure
2.3.2 Our multiphysical approach
2.4 A NEW MODEL TO SUPPORT DECISION MAKING DURING THE DESIGN OF COMPOSITE STRUCTURES
2.4.1 Fast simulation for design choices validation
2.4.2 Interactive exploration of design solution spaces from the PRBM
2.4.3 Our approach for composite structure design
2.5 QUALIFICATION
2.6 CONCLUSION
CHAPTER 3 PARAMETRIZATION OF A MODEL:
3.1 INTRODUCTION
3.2 DETAILS OF THE PRBM MODEL
3.2.1 Properties
3.2.2 Definition of the PRBM
3.3 DETAILS AND PRINCIPLES OF THE PGD METHOD
3.4 MODEL SEPARATION WITH PGD
3.4.1 Basis of modeling
3.4.2 Parametrization with PGD
3.5 SIMULATION OF A LAMINATED COMPOSITE BEAM: THE USUAL APPROACH
3.5.1 Details of the case of study
3.5.2 FEM models
3.5.3 Basis: FEM simulation
3.6 PRBM BASED SIMULATION
3.6.1 Details about the PGD approach
3.6.2 PRBM results in the laminated composite beam
3.6.3 Qualification of the PRBM for static behavior
3.7 CONCLUSION
CHAPTER 4 MULTIPHYSICAL MODELING:
4.1 INTRODUCTION
4.2 VISCOELASTIC BEHAVIOR IN A LAMINATED COMPOSITE STRUCTURE
4.2.1 Experimental approach
4.2.2 Demonstration of viscoelasticity
4.3 PRBM CONSTRUCTION FOR A MULTIPHYSICAL APPROACH.
4.4 FRACTIONAL DERIVATIVE TO MODEL VISCOELASTICITY
4.4.1 Modeling of creeping
4.4.2 Modeling viscoelasticity in the PRBM
4.4.3 Determination of the fractional parameters from experimentation ..
4.5 PRBM BASED SIMULATION OF DYNAMIC BEHAVIOR IN A COMPOSITE STRUCTURE
4.5.1 Study case
4.5.2 A separated and reduced model of the dynamic behavior of the composite beam
4.5.3 Model processing for CAE
4.6 CONCLUSION
CHAPTER 5 THE PRBM SUPPORTING DECISION MAKING IN THE DESIGN OF COMPOSITE STRUCTURES
5.1 INTRODUCTION
5.2 PRBM FOR DESIGN
5.3 A KNOWLEDGE MODEL TO DESIGN
5.3.1 The parametric knowledge model
5.3.2 Modeling of design objectives
5.3.3 Modelling of design constraints
5.4 OPTIMIZATION: A STRATEGY TO EXPLORE DESIGN SPACES INTERACTIVELY
5.4.1 The choice of evolutionary approach
5.4.2 From the PKM to the optimization model
5.5 MODELING A DESIGN PROBLEM FOR AG PROCESSING
5.5.1 Representation of the chromosome
5.5.2 Description of the optimization process.
5.6 A DESIGN PROBLEM SOLVING
5.6.1 Design problem details: a laminated composite plate
5.6.2 Design solutions after optimization
5.6.3 Design of laminated composite structures from the PKM
5.7 QUALIFICATION
5.7.1 Performance of the optimization process
5.7.2 Qualification of the design solution having static behavior: validation from a FEM simulation
5.7.3 Qualification of the design solution having dynamic behavior: validation from experiments
5.8 CONCLUSION
CHAPTER 6 CONCLUSIONS AND PERSPECTIVES
6.1 CONCLUSIONS
6.2 PERSPECTIVES
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REFERENCES
