## Thesis etd-12112009-125407 |

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Thesis type

Tesi di dottorato di ricerca

Author

CETTA, FABIO

email address

fabio_cetta@libero.it

URN

etd-12112009-125407

Thesis title

Design Methodologies of Aeronautical Structures
with Acoustic Constraints

Academic discipline

ING-IND/04

Course of study

INGEGNERIA AEROSPAZIALE

Supervisors

**correlatore**Morino, Luigi

**tutor**Prof. Frediani, Aldo

Keywords

- Finite Elements
- Hermite interpolation
- interior acoustics
- MDO

Graduation session start date

21/12/2009

Availability

Full

Summary

The aim of this thesis is to present a finite element methodology, based upon a three-dimensional extensions of the classical Hermite interpolation and of the Coons Patch, for the evaluation of the natural modes of vibration of the air inside cavities (interior acoustics) and of elastic structures (structural dynamics). This methodology is thought for acoustic applications within Multidisciplinary Desing and Optimization, where computational effectiveness is a key attribute, especially during iterative optimization.

The distinguishing feature of the proposed technique is its high efficiency,

with the possibility to capture relatively high spatial frequency modes (essential in acoustics) even using a limited number of degrees of freedom. Also, the element is quite flexible and may be used for modeling any three-dimensional geometry. For instance, thin-wall structures like shells and plates are treated with three-dimensional brick elements with a single element along the thickness. An additional advantage is related to the possibility of applying a quasi-static reduction, which allows one to eliminate those degrees of freedom associated with the derivatives while maintaining a high level of accuracy, so that to further improve the effectiveness of the element.

The classical one-dimensional Hermite interpolation is an interpolating technique of order three that uses the function and its derivative at the end points of the element. The classical Hermite technique for one-dimensional domains can be extended to higher orders, by including higher derivatives as nodal unknowns, thereby increasing the class of the element. Then, the three-dimensional extension is obtained combining the Hermite polynomials in each direction. For example, in the three-dimensional third-order interpolation, the unknowns are the nodal values of the function, of its three partial derivatives, of its three mixed second derivatives, and of its third mixed derivative. Similarly to the one-dimensional approach, higher orders are then obtained by including higher derivatives at nodes.

The Hermite element, even that of order three, is rarely used because of problems that arise whenever the domain is not topologically hexahedral, that is when the coordinate

lines (and so the base vectors) of two adjacent blocks present a discontinuity. Specifically, as far as the first-order derivatives are

concerned, the problem has been removed by assuming as unknowns the Cartesian coordinates of the gradient, since they are continuous across block boundaries. The problem remains for the higher-order derivatives: in order to express them in terms of Cartesian components, their set should be complete (in particular, we have only the mixed second derivatives and, hence, incomplete information on the Hessian matrix). The remedies to this issue are key features of the present thesis. In particular, two solutions have been proposed in this thesis: (1) the high-order derivatives relative to different blocks are treated as independent unknowns at the block boundaries; (2) a new 3-D high-order internal-nodes family of elements based upon the Coons Patch is used: these elements are defined so as to have only the function and the three derivatives as nodal unknowns (thereby, the higher the order of the element the higher the quantity of internal nodes needed for the interpolation).

To be specific, the Coons Patch pertain the interpolation over a

quadrilateral surface. Given the four edge lines, the Coons Patch

is obtained as the sum of the two linear interpolations

between opposite boundary lines, minus a bilinear

interpolation through the four corner points. From this technique stems the idea of a new family of elements, which edges are generated using the Hermite interpolation. The objective is to extend the use of high-order elements based on a Hermite approach also to generically complicated geometries, so as to take advantage of their effectiveness. These elements will be referred to as Hybrid elements.

The validation is based upon the evaluation of the natural eigenvalues (or natural frequencies) and modes of the air vibrating inside hexahedral cavities as well as of those of elastic thin plates (for each of this case exact or accurate solutions are available).

Applications to quite complicated structures, such as curved domains (cylindrical cavities) or very simplified wing-boxes are presented. The results are compared with those obtained using commercial softwares (such as Ansys).

Comparisons with the literature are also included.

The distinguishing feature of the proposed technique is its high efficiency,

with the possibility to capture relatively high spatial frequency modes (essential in acoustics) even using a limited number of degrees of freedom. Also, the element is quite flexible and may be used for modeling any three-dimensional geometry. For instance, thin-wall structures like shells and plates are treated with three-dimensional brick elements with a single element along the thickness. An additional advantage is related to the possibility of applying a quasi-static reduction, which allows one to eliminate those degrees of freedom associated with the derivatives while maintaining a high level of accuracy, so that to further improve the effectiveness of the element.

The classical one-dimensional Hermite interpolation is an interpolating technique of order three that uses the function and its derivative at the end points of the element. The classical Hermite technique for one-dimensional domains can be extended to higher orders, by including higher derivatives as nodal unknowns, thereby increasing the class of the element. Then, the three-dimensional extension is obtained combining the Hermite polynomials in each direction. For example, in the three-dimensional third-order interpolation, the unknowns are the nodal values of the function, of its three partial derivatives, of its three mixed second derivatives, and of its third mixed derivative. Similarly to the one-dimensional approach, higher orders are then obtained by including higher derivatives at nodes.

The Hermite element, even that of order three, is rarely used because of problems that arise whenever the domain is not topologically hexahedral, that is when the coordinate

lines (and so the base vectors) of two adjacent blocks present a discontinuity. Specifically, as far as the first-order derivatives are

concerned, the problem has been removed by assuming as unknowns the Cartesian coordinates of the gradient, since they are continuous across block boundaries. The problem remains for the higher-order derivatives: in order to express them in terms of Cartesian components, their set should be complete (in particular, we have only the mixed second derivatives and, hence, incomplete information on the Hessian matrix). The remedies to this issue are key features of the present thesis. In particular, two solutions have been proposed in this thesis: (1) the high-order derivatives relative to different blocks are treated as independent unknowns at the block boundaries; (2) a new 3-D high-order internal-nodes family of elements based upon the Coons Patch is used: these elements are defined so as to have only the function and the three derivatives as nodal unknowns (thereby, the higher the order of the element the higher the quantity of internal nodes needed for the interpolation).

To be specific, the Coons Patch pertain the interpolation over a

quadrilateral surface. Given the four edge lines, the Coons Patch

is obtained as the sum of the two linear interpolations

between opposite boundary lines, minus a bilinear

interpolation through the four corner points. From this technique stems the idea of a new family of elements, which edges are generated using the Hermite interpolation. The objective is to extend the use of high-order elements based on a Hermite approach also to generically complicated geometries, so as to take advantage of their effectiveness. These elements will be referred to as Hybrid elements.

The validation is based upon the evaluation of the natural eigenvalues (or natural frequencies) and modes of the air vibrating inside hexahedral cavities as well as of those of elastic thin plates (for each of this case exact or accurate solutions are available).

Applications to quite complicated structures, such as curved domains (cylindrical cavities) or very simplified wing-boxes are presented. The results are compared with those obtained using commercial softwares (such as Ansys).

Comparisons with the literature are also included.

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