International Journal of Aerospace Engineering

Volume 2017 (2017), Article ID 7293682, 20 pages

https://doi.org/10.1155/2017/7293682

## A Novel Highly Accurate Finite-Element Family

Department of Engineering, Roma Tre University, Via della Vasca Navale 79, 00146 Rome, Italy

Correspondence should be addressed to Giovanni Bernardini

Received 7 February 2017; Revised 14 June 2017; Accepted 5 July 2017; Published 5 September 2017

Academic Editor: Christopher J. Damaren

Copyright © 2017 Giovanni Bernardini et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A novel th order finite element for interior acoustics and structural dynamics is presented, with arbitrarily large. The element is based upon a three-dimensional extension of the Coons patch technique, which combines high-order Lagrange and Hermite interpolation schemes. Numerical applications are presented, which include the evaluation of the natural frequencies and modes of vibration of (1) air inside a cavity (interior acoustics) and (2) finite-thickness beams and plates (structural dynamics). The numerical results presented are assessed through a comparison with analytical and numerical results. They show that the proposed methodology is highly accurate. The main advantages however are (1) its flexibility in obtaining different level of accuracy (-convergence) simply by increasing the number of nodes, as one would do for -convergence, (2) the applicability to arbitrarily complex configurations, and (3) the ability to treat beam- and shell-like structures as three-dimensional small-thickness elements.

#### 1. Introduction

Interior cabin noise is a challenging problem in most aircraft design and has received considerable attention in the last decade. This is particularly true for rotary wing aircraft because the propulsive system induces direct acoustic disturbances and fuselage vibrations that, in turn, may cause unacceptable ride discomfort inside the cabin area hosting passengers, as well as significant impact on the fatigue-life of the structures and hence on the maintenance costs [1–3]. One of the main characteristics of aircraft cabin noise is the wide range of frequencies of interest, due to different noise sources: fuselage boundary-layer, airborne, and structure-borne noise are among the most important ones [4]. Boundary-layer noise, generated by the shaking of the fuselage-wall due to external turbulence pressure fluctuations, is a random, broadband, high-frequency signal. Airborne noise is a mid/low-frequency tonal noise associated with periodic pressure fluctuations from the propulsive system impinging the fuselage structure that, in turn, excites interior acoustics, whereas structure-borne noise is related to the acoustic energy generated by periodic vibratory loads (rotor hub-loads, engine vibrations, gearbox, etc.) acting on the airframe.

From the above considerations, one may infer that interior noise prediction is a challenging problem that requires the development of efficient formulations able to model accurately the interactions between the fuselage structural dynamics and the cabin interior acoustics, even in the presence of wide frequency band signals. An approach that may be used to address such a problem consists of coupling Finite-Element Methods (FEM) for structural dynamics with Boundary Element Methods (BEM) for interior acoustics [5]. However, in several practical applications both standard FEM and BEM methods become inaccurate and computationally intensive when the number of elements needed to model the problem becomes large [6]. In fact, the accuracy of these methods strictly depends on both the number of nodes used in the model and the order of the shape functions used to interpolate the solution within each element: a rule of thumb to maintain accuracy in both methods is to have six linear or three parabolic elements per wavelength [6]. Moreover, in several practical applications FEM are more efficient than BEM for the numerical solution of interior acoustic problems [7].

In the last few years, considerable work has been done to improve the prediction capabilities of FEM algorithms in analyzing structural dynamics/interior acoustics problems characterized by midfrequency signals [8, 9]. Among these, we can mention the Galerkin least-squares finite-element method for the solution of the two-dimensional Helmholtz equation [10], the Galerkin residual-free bubbles method [11, 12], the smoothed FEM using cubic spline polynomial functions in hexahedral elements [13], and the isogeometric FEM [14]. During the years, very accurate FEM methods suited for structural dynamics and fluid dynamics applications have been developed; among these, it is worth mentioning the Spectral Finite-Element Method (SFEM), based on Lagrange polynomials on the Gauss–Lobatto–Legendre grid [15], the Fourier transform-based and Wavelet transform-based spectral FEM [16, 17], the hierarchical -FEM [18–20], and -FEM methods based upon isoparametric Hermite elements [21, 22]. Since its origin, SFEM have been successfully applied in several fields of the physics and applied sciences: acoustics, fluid dynamics, heat transfer, and structural dynamics. In mechanical/aeronautical engineering they are widely used in wave propagation, in both homogeneous and inhomogeneous structures [15, 23], in structural health monitoring applications [24], as well as in structural dynamics of rotating beams [25].

In this paper, we concentrate on finite elements. This methodology is widely used in engineering and science applications [26, 27] and is extensively analyzed from both practical and theoretical points of view [20, 28–30]. Specifically, we propose a new efficient and accurate finite-element methodology, suited for the analysis of both structural dynamics and interior acoustics. The formulation proposed here is a user-friendly evolution of the methodology developed in the past by the authors and their collaborators. This was presented in [31–38], with increasing level of algorithmic sophistication. This work is to be understood as a first step towards the development of a highly accurate finite-element formulation for the evaluation of the natural modes of vibration of the air inside a cavity (interior acoustics) and/or of an elastic structure (structural dynamics), for relatively high spatial frequencies (specifically higher than those that may be efficiently obtained with the methodologies presently available), so as to make the technique useful for structural acoustics applications, which involve the coupling of structure and air. Specifically, the finite-element methodology presented here is based upon a combination of two important techniques: () the three-dimensional extension of the Coons patch technique [39–41] and () the high-order Lagrange and Hermite interpolation schemes [42–44]. This combination is very powerful and yields the distinguishing feature of combining high efficiency (with the possibility of capturing relatively high spatial frequencies, as required in aeroacoustics applications), with user-friendliness, giving a finite element that is very accurate and computationally efficient. More interestingly, it has the unique feature of being flexible, in the sense that one may increase the order of the scheme accuracy (-convergence) just by changing the number of nodes (as one would for -convergence). Another important feature is its applicability to arbitrarily complex configurations, using always the same type of element. In particular, the formulation covers both interior acoustics and structures and is able to model beams and plates, which are to be treated as three-dimensional small-thickness structures. The above characteristics are important in that the technique proposed here is envisioned within the context of fully automated multidisciplinary optimal design [45]. In addition the following two features (not all simultaneously present in other finite elements currently available for aeroacoustics) appear important: () the element captures modes with relatively high spatial frequencies (as essential for aeroacoustics utilization) and does so efficiently, so as to give good results with relatively few elements, an important feature when repeated calculations occur, as in automated optimization, and () the element is user-friendly.

#### 2. Preliminaries: Lagrange and Hermite Interpolation

The formulation presented here involves a judicious combination of high-order Lagrange and Hermite interpolation polynomials [42–44], which are briefly reviewed here. First, we address the two-point interpolation polynomials (needed in Sections 3–5), and then we consider the -point interpolation, with arbitrarily large (used in Section 6).

*The Two-Point Lagrange and Hermite Interpolation Polynomials*. Here, we discuss the two-point Lagrange and Hermite interpolation polynomials [42–44]. Consider a function , defined over the interval . Let us divide the interval into subintervals, for which we introduce a local coordinate , so as to have that the end points of each subinterval are given by . Within each subinterval, the Lagrange linear interpolation is given by where denotes the values of at the end points, whereas the Lagrange first-order interpolation polynomials are given by The resulting interpolated function over the whole interval is continuous (class ) and piecewise linear. It will be referred to as the* one-dimensional first-order Lagrange interpolation*.

On the other hand, within each subinterval, the standard Hermite interpolation is where denotes the values of at and denotes the values at , whereas the Hermite interpolation polynomials and are given by The resulting interpolated function in is continuous with its first derivative (class ) and is piecewise cubic. It will be referred to as the* one-dimensional third-order Hermite interpolation*.

*The *-*Node Lagrange and Hermite Interpolation Polynomials*. Here, we review the -node Lagrange and Hermite interpolation polynomials [42, 44].

Let us consider first the -node Lagrange interpolation; namely, where , is the total number of subintervals between the nodes (), and [Throughout the paper and coincide with the end points of the interval under consideration.]

In particular, if , , and , we have , in agreement with (2). Note that the Lagrange interpolation polynomials satisfy the standard interpolation condition

It is well known that, for , the Lagrange interpolation polynomials are unstable if are uniformly spaced [44]. However, the instability disappears if coincide with the Gauss quadrature abscissas. Unfortunately, the Gauss quadrature abscissas do not include the end points of the interval. The Gauss–Lobatto quadrature scheme on the other hand includes these points and is not affected by the above instability issue. Accordingly, in the case of high-accuracy schemes, used in this paper coincide with the Gauss–Lobatto abscissas.

Next, let us consider the -node extension of the two-point Hermite interpolation presented in (3) and (4). There exist two possible approaches. The first is to increase the order of the derivatives at the end points. The other consists in using as interpolation parameters the function and its first derivative not only at the end points but also at additional interior points. As shown in [36], the latter is the most convenient (and definitely the most user-friendly). Accordingly, here we concentrate on such an approach.

In analogy with the third-order formulation (see (3)), we have where the polynomials and are given by [43] with obtained by imposing These polynomials satisfy the Hermite interpolation conditions Indeed, it is apparent that the polynomials and vanish with their first derivatives at , for all (see (8)). Thus, we only have to verify what happens when . It is apparent that . In addition, taking the logarithmic derivative of , we have which yields, (see (11)). Moreover, it is apparent that . In addition, we have which yields (see (8)). In summary, and satisfy all the conditions in (12).

The polynomials have degree . We will refer to this as the interpolation of order .

#### 3. Motivation: Hermite Brick and BVD Problem

In order to set this work in the proper context, we present some details of the formulation for the so-called* Hermite brick*. This allows us to motivate the introduction of the family of high-order elements proposed here.

*Third-Order Hermite Brick*. In order to place the finite-element family proposed here in the proper perspective, let us consider a drawback of the Hermite interpolation. To this end, assume that the problem is defined in a topologically hexahedral region. Let us subdivide the region into topologically hexahedral subregions, here referred to as the* brick*, which are described by the mapping where () are curvilinear coordinates, whereas is a suitably smooth function. [Sometimes it is more convenient to use the symbols (); other times we prefer to use . Accordingly, we use and interchangeably.]

In this section, we use the third-order Hermite interpolation in all directions (for both geometry and unknown function).

This yields the following interpolation for the unknown ; namely, The symbol , with , where stands for either + or −, is understood as a sum that spans over the eight values of , which correspond to the eight vertices of the brick. Furthermore, the symbol denotes the partial derivative with respect to : . Similarly, we have and . In addition, note that the Hermite interpolation scheme requires only the nodal values of the function and its first partial derivatives in each direction. No second repeated derivative with respect to the same variable arises. Accordingly, the second-derivative summation spans only over the mixed derivatives, and hence . Moreover, the term 123 is the only mixed third-order derivative. Finally, , , , and are suitable products of the Hermite polynomials in (4). For instance,

In summary, for the three-dimensional third-order Hermite interpolation scheme, the finite-element unknowns are the nodal values of () the unknown function, () its three first-order partial derivatives, () its three second-order mixed derivatives, and () its third-order mixed derivative . This yields a total of eight unknowns per node, out of which only one is the nodal value of the function, the other seven being related to the various derivatives. The finite element described above will be referred to as the* third-order Hermite brick*.

The expressions in (17) provide the local finite-element shape functions. They may be assembled to yield a global finite-element interpolation over the whole block, of the type where the unknowns comprise the values of and of all the partial derivatives mentioned above, evaluated at the nodes. On the other hand, are the global shape functions [20], obtained by assembling the local shape functions in (17).

*The Base Vector Discontinuity (BVD) Problem*. The third-order Hermite brick is seldom used, because of a problem that may arise at the interface of two or more bricks. Specifically, the issue arises when the coordinate lines of two adjacent bricks present a discontinuity, namely, when the covariant base vectors which are tangent to the coordinate lines, are discontinuous (either in magnitude or direction). In this case, the partial derivatives of a function with respect to are given by and are discontinuous. As far as the first-order derivatives are concerned, the problem is removed by assuming as unknowns the values of the Cartesian coordinates of . The first-order partial derivatives may then be obtained using (20).

The problem, however, remains for the second-order derivatives, because, in order to express them in terms of Cartesian components, one needs the complete Hessian matrix, namely, all the second-order derivatives, and not only the mixed ones, which are the only ones utilized in the three-dimensional Hermite interpolation. Similar considerations hold for the third-order derivatives. The problem discussed above will be referred to as the* BVD problem* (Base Vector Discontinuity problem). [It may be noted that the Hermite scheme is not limited to the third order. Higher order schemes may be easily introduced. However, these extensions are also subject to the BVD problem, and hence of little interest here.]

In order to remedy the problem, various unsatisfactory attempts have been considered by the authors and their collaborators and are summarized in [31, 32]. The problem was resolved in [33–38] with the introduction of the Coons patches [39–41]. The corresponding approach is addressed here, along with recent developments.

#### 4. The Coons Patch and Its Three-Dimensional Extension

Throughout the rest of this paper, the region of definition of the problem under consideration consists of the union of topologically hexahedral domains, referred to as* blocks*, which, in analogy with (15), are described by the mapping [This allows for sufficient generality, since this subdivision may be easily obtained for any region of interest in practical applications.] Then, the intervals are divided into subintervals (not necessarily uniform, in analogy with (5)). By doing this, each domain (block) has been subdivided into subdomains called* bricks*, which are described by (15). In the rest of the paper, the term* “element”* is identified with the term* “block*.*”* However, a block may consist of a single brick (single-brick element).

The various formulations presented are assumed to be* isoparametric*, in the sense that the same type of interpolation scheme is used for the geometry and the unknown.

The finite-element family proposed here, which, as we will see, is not affected by the BVD problem, is based upon an approach introduced by Coons [39–41], for approximating a quadrilateral surface in terms of its four edges, namely, generating a suitable quadrilateral surface of which we know only its four edges. The resulting surface is known as the* Coons patch* and is illustrated here, along with its three-dimensional extension.

The Coons patch technique is also known as the* transfinite interpolation*, introduced in [46, 47].

*The Coons Patch*. The Coons patch was introduced to describe geometries. Accordingly, the scheme is discussed in relationship with the geometry, which is also less cumbersome to describe. This yields no loss of generality, because, as mentioned above, we use an isoparametric formulation.

Let , with , describe a generic topologically quadrilateral surface patch (see Figure 1). Let be the equations that describe the four edges of the patch, and let denote the four corner points.