# Optimal Design of Plated/Shell Structures under Flutter Constraints—A Literature Review

^{*}

## Abstract

**:**

**Contents**

**1.****Introduction****2.****Brief Description of the Flutter Problem****3.****Remarks on the Formulation of Optimization Problems****4.****Objective Functions**- 4.1.
- Deterministic Approach
- 4.2.
- Reliability Analysis
**5.****Physical (Material) Design Variables**- 5.1.
- Composite Materials
- 5.2.
- Functionally Graded Materials and Nanocomposite Structures—Thermal Protection
- 5.3.
- Piezoelectric (PZT) Patches–Active and Passive Flutter Control
- 5.4.
- Sandwich Structures
**6.****Geometric Design Variables**- 6.1.
- Cross-Section Parameters–Variable (Stepped) Thickness
- 6.2.
- Form of the Structure
**7.****Numerical (Finite Element) Packages****8.****Optimization Algorithms****9.****Concluding Remarks**

## 1. Introduction

## 2. Brief Description of the Flutter Problem

_{1}, L

_{2}are linear differential operators with respect to x, y variables, w denotes the normal to the mid-surface displacements, t is a time and h means the thickness. For the Love-Kirchhoff kinematical hypothesis the explicit form of the differential operators is presented by Bolotin [7,8], Hedgepeth [9], Houbolt [10], Sawyer [11], Bohon [12], Stepanov [13], Muc, Flis [14].

_{1}, L

_{2}but not their order—see e.g., Li, Song [16].

## 3. Remarks on the Formulation of Optimization Problems

- A vector of design variables and a space of design variables;
- An objective function or an objective functional;
- A set of constraints in the form of equality or inequality.

_{i}can be represented as:

_{i}∈ R, i = 1, 2, …, I

_{i}∈ C = {c

_{1},c

_{2}…}, i = 1, 2, …, I; c

_{p}∈ R or c

_{p}∈ N

- In dimensional (parametric) optimization, design variables determine the structure thickness distribution and its parameters characterizing the cross-section;
- In shape optimization, these are the describing variables:
- i.
- The geometry (and thus also the shape) of the outer edge of the structure;
- ii.
- The geometry of the mid-surface of the structures (beams, plates, shells).
- iii.
- In the optimization of the topological structure, the design variables define:
- iv.
- The manner of connection of elements, areas, or components of the structure;
- v.
- The number and spatial distribution of structure elements;
- vi.
- Material distribution in the structure.

- Elementary cell,
- Individual layer,
- Laminate.

- Physical (material) representing the CM structure from which the structure is made;
- Geometric—characterizing the geometry of the structure.

- Only by selecting the material when looking for the optimal distribution of the laminate thickness, the thickness and shape of the reinforcing patches, the shape of the middle surface of the structure, or the shape of the edge;
- By designing a new material, if in the above problems there are additional constraints (technological, geometric, etc.,) that none of the currently available materials meet.

## 4. Objective Functions

#### 4.1. Deterministic Approach

- The direct formulation of the problem (Muc, Flis [14]):

- The implicit formulation with a bound (Song, Li, Carrera, Hagedorn [29]):

- The implicit formulation with a bound (Guo [30]):

- The maximization of weighted sum of the critical aerodynamic pressures under different probability density function of flow orientations (Li, Narita [31]):

- A minimum weight wing W subject to divergence/flutter constraints (Kameyama, Fukunaga [34])

- Maximization of the flutter critical parameter Q
_{crit}, i.e., a function of the panel’s stiffnesses, damping, and dynamic pressure of the free-stream. (Vijay, Durvasulah [35])

- The uncertainty problem—the minimization of the additional masses w
_{i}added to the wing construction and satisfying the frequency constraint (Kuttenkeuler, Ringertz [36])

#### 4.2. Reliability Analysis

_{crit}is the objective function of the optimization (flutter speed, eigenfrequency, weight–see (Equations (11)–(17)), and f

_{e}means the design value of the objective function. For the sake of the simplicity of the aeroelastic analysis the problem (19) is usually rewritten as the MinMax or MaxMin problem (see Muc, Kędziora [38]) referring to the graphical representation plotted in Figure 3.

## 5. Physical (Material) Design Variables

#### 5.1. Composite Materials

- Constant as $\left(x,y\right)\in \mathsf{\Omega}$
- Constant as $\left(x,y\right)\in {\mathsf{\Omega}}_{i},{\mathrm{U}}_{i}{\mathsf{\Omega}}_{i}=\mathsf{\Omega},i1$
- Variable as $\left(x,y\right)\in \mathsf{\Omega}$

_{i}in the case shown in Figure 4b. The number of design variables increases for arbitrary orientations in each of the plies constituting laminate. For the Love-Kirchhoff hypothesis the laminate configuration is represented by 12 lamination parameters in the domain Ω (or each of the domains Ω

_{i}). It grows further for transverse shear deformation theories—see Muc [56].

_{11}, A

_{12}, A

_{22}, A

_{66}, A

_{16}, A

_{26}, and D

_{11}, D

_{12}, D

_{22}, D

_{66}, D

_{16}, D

_{26}—see Muc [57]. However, they are not independent and can be expressed by four natural values: N

_{0}—the number of pair of plies oriented at 0

^{0}, N

_{90}—the number of pair of plies oriented at 90

^{0}and:

_{0}, N

_{90}although the mapping is not unique—Muc [58]. Therefore, the space (plane) ${N}_{0}^{D}$, ${N}_{90}^{D}$ seems to be the most convenient representation of the optimization results—Figure 5. To compare discrete with angle-ply (continuous) fiber orientations it is better to introduce the following definition:

_{11}is observed. Vijay, Durvasulah [35] introduced the critical parameter (17) as the value divided by the D

_{11}value. The similar effects were noticed by Muc, Flis [14], Rikards, Teters [15]—see Figure 6. It is interesting to note that the decrease of the aerodynamic critical pressure with the growth of the fiber orientations (the angle-ply configuration is considered) is almost insensitive to the form of boundary condition and the orthotropy ratio E

_{L}/E

_{T}.

#### 5.2. Functionally Graded Materials and Nanocomposite Structures—Thermal Protection

- 1.
- Ceramic/metal (FGM) structures with ceramic (C) and metal (M) isotropic properties and the prescribed form of a grading function having an unknown power law coefficient n.$${E}_{FGM}={E}_{m}\left(1-{V}_{C}\right)+{E}_{C}{V}_{C},{V}_{C}={\left(\frac{z}{t}+\frac{1}{2}\right)}^{n},0\le n<\infty ,-\frac{t}{2}\le z\le \frac{t}{2}$$
- 2.
- Sandwich structures made of a FGM core and laminated faces; the core Young’s modulus can be determined from the following relation:$${E}_{FGM}={E}_{s}\left[{\left(1-\frac{{\rho}_{FGM}^{}}{{\rho}_{s}^{}}\right)}^{2}\frac{{\rho}_{FGM}^{2}}{{\rho}_{s}^{2}}+\frac{{\rho}_{FGM}^{2}}{{\rho}_{s}^{2}}\right]$$
_{FGM}and ρ_{FGM}(grading function) are the Young’s modulus and mass density of the core, respectively; E_{s}and ρ_{s}are the Young’s modulus and mass density of the solid material. - 3.
- Carbon nanotubes (CNT) embedded in the matrix–orthotropic properties of CNTs (four material constants), Young’s modulus of the matrix, and the volume fraction and distribution of CNTs (Ref [118]).

#### 5.3. Piezoelectric (PZT) Patches–Active and Passive Flutter Control

#### 5.4. Sandwich Structures

## 6. Geometric Design Variables

#### 6.1. Cross-Section Parameters–Variable (Stepped) Thickness

#### 6.2. Form of the Structure

## 7. Numerical (Finite Element) Packages

- To introduce different variants of boundary conditions;
- To investigate arbitrary laminate configurations with no elimination of the B
_{ij}, A_{16}, A_{26}, D_{16}, D_{26}terms in the stiffness matrices; - To use the first order transverse shear shell/plate theories instead of the simplest Love-Kirchhoff theories.

- The problems with the accuracy of solved flutter problems;
- The problems with the solution of the optimization problems, particularly for laminated structures where non-uniqueness of solutions is commonly encountered.

## 8. Optimization Algorithms

## 9. Concluding Remarks

- 1.
- The majority of considerations is based on the parametric investigations by observing the influence of various effects on values of objective functions; it is especially visible in problems dealing with geometric design variables;
- 2.
- The optimization algorithms (evolutionary techniques) are mainly employed in three groups of problems:
- a.
- Searching for the optimal stacking sequences in laminated structures or sandwich structures with laminated facesheets; in the paper a special attention is focused on the reduction of the total number of design variables for multilayered laminate constructions;
- b.
- Location and final shapes of piezoelectric actuators/sensors used in the active or passive control of the structural response;
- c.
- Variable thickness optimization of structures.

- 3.
- The broader use of optimization methods is, in our opinion, required in the following class of problems:
- a.
- Shape optimization of structures considered, particularly in view of their loss of dynamic stability (geometric design variables);
- b.
- Topology optimization of grading functions introduced for ceramic/metal (functionally graded materials) and/or nanocomposites reinforced with carbon nanostructures

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Variations of the natural frequencies with aerodynamic pressure. (

**a**) Isotropic plates; (

**b**) multilayered laminated plates.

**Figure 2.**Schematical representation of the pre-flutter (free vibrations) and post-flutter modes (1D cross-section).

**Figure 4.**Possible constructions of the individual layers made of 2D fibers: (

**a**) plain fibers, (

**b**) plain fibers in the fragment of the structure, (

**c**) curvilinear fibers.

**Figure 6.**Variations of the normalized critical pressures for square angle-ply plates (L

_{x}/L

_{y}= 1, E

_{L}/E

_{T}= 40).

Type | Structural Theory | Aerodynamic Theory | Mach Number M |
---|---|---|---|

1 | Linear | Linear piston theory | $\sqrt{2}\le \mathrm{M}\le $5 |

2 | Linear | Linear potential theory | $1\le \mathrm{M}\le $5 |

3 | Nonlinear | Linear piston theory | $\sqrt{2}\le \mathrm{M}\le $5 |

4 | Nonlinear | Linear potential theory | $1\le \mathrm{M}\le $5 |

5 | Nonlinear | Nonlinear piston theory | M > 5 |

6 | Nonlinear | Navier-Stokes equations | Transonic, supersonic, hypersonic |

Multilayered Composite Laminates | FGM | Nanocomposites | PZT | Sandwich | ||
---|---|---|---|---|---|---|

Angle-Ply | Discrete 0/45/90 | Curvilinear Fiber Format | ||||

1 | 4 | Parameters defining the characteristic curve | Mechanical properties of the constituents Parameters defining the grading function | Voltage Positions of the patches | Mechanical properties of the core |

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**MDPI and ACS Style**

Muc, A.; Flis, J.; Augustyn, M.
Optimal Design of Plated/Shell Structures under Flutter Constraints—A Literature Review. *Materials* **2019**, *12*, 4215.
https://doi.org/10.3390/ma12244215

**AMA Style**

Muc A, Flis J, Augustyn M.
Optimal Design of Plated/Shell Structures under Flutter Constraints—A Literature Review. *Materials*. 2019; 12(24):4215.
https://doi.org/10.3390/ma12244215

**Chicago/Turabian Style**

Muc, Aleksander, Justyna Flis, and Marcin Augustyn.
2019. "Optimal Design of Plated/Shell Structures under Flutter Constraints—A Literature Review" *Materials* 12, no. 24: 4215.
https://doi.org/10.3390/ma12244215