Advanced Vibroacoustic Simulations Using Isogeometric Analysis ^†

Abstract

This work presents a methodology for integrating Computer-Aided Design (CAD), specifically using Rhinoceros 6, with Isogeometric Analysis (IGA) for vibroacoustic simulations. Since CAD models typically provide only boundary representations, the 3D domain is reconstructed through an immersed IGA approach. The methodology is first illustrated in a 1D setting and then extended to a 3D case. The vibroacoustic coupling strategy is also described, enabling an efficient analysis of coupled fluid–structure vibrations. The proposed framework ensures direct CAD/CAE integration, thereby reducing preprocessing efforts.

1. Introduction

Vibroacoustic problems, namely the interaction between elastic structures and surrounding fluids, play a central role in several engineering fields such as automotive, aerospace, and architectural acoustics [1]. Accurate modelling of these phenomena is crucial for predicting and controlling noise transmission and unwanted vibrations, which directly affect passenger comfort and structural performance [1,2]. Classical numerical techniques, such as the Finite Element Method (FEM) and the Boundary Element Method (BEM), are widely employed; however, they exhibit well-known limitations, particularly when large-scale problems lead to very high numbers of degrees of freedom [3].

Isogeometric Analysis (IGA) was introduced to address these challenges by exploiting the same spline functions, typically B-splines and Non-Uniform Rational B-Splines (NURBS), used in CAD systems to represent geometries, as well as to approximate the unknown physical fields [4]. This feature enables an exact description of the geometry, improved continuity of the basis functions, and superior convergence properties compared to traditional FEM [5]. In vibroacoustic applications, IGA has proven capable of providing more accurate predictions of eigenfrequencies, frequency responses, and fluid–structure coupling effects [5,6].

A major obstacle, however, remains the integration between CAD and numerical analysis. CAD models generally provide only a boundary representation (B-rep), whereas numerical simulations require a full volumetric description of the domain [7]. To overcome this issue, the immersed IGA technique has been developed, enabling the construction of three-dimensional IGA meshes directly from boundary representations [7,8,9,10,11]. This approach significantly reduces preprocessing costs, simplifies the handling of complex geometries, and preserves the accuracy and continuity advantages of IGA.

While this methodology has already been applied in the literature to the construction of 3D structural and optimization problems [12,13], the originality of the present work lies in the extraction of a 3D acoustic cavity via immersed IGA and its subsequent coupling with an external structure.

The proposed methodology is first illustrated through a 1D case to highlight the fundamental principles, and it is then extended to a 3D application, demonstrating its effectiveness and potential for engineering vibroacoustic applications.

2. Materials and Methods

2.1. Vibroacoustic Matrix Formulation

Let us introduce the vibroacoustic matrix formulation. The problem can be expressed in matrix form according to the following operators:

$$\begin{array}{cccc}\hfill k(\delta u^h,u^h)& =\delta {\mathbf{u}}^T\mathbb{K}\mathbf{u}\hfill & \hfill & \Rightarrow \mathbb{K}:\mathrm{Structural}\mathrm{Stiffness}\mathrm{Matrix}\hfill \end{array}$$

(1)

$$\begin{array}{cccc}\hfill m(\delta u,u)& =\delta {\mathbf{u}}^T\mathbb{M}\mathbf{u}\hfill & \hfill & \Rightarrow \mathbb{M}:\mathrm{Structural}\mathrm{Mass}\mathrm{Matrix}\hfill \end{array}$$

(2)

$$\begin{array}{cccc}\hfill c(\delta u,p)& =\delta {\mathbf{u}}^T\mathbb{C}\mathbf{p}\hfill & \hfill & \Rightarrow \mathbb{C}:\mathrm{Coupling}\mathrm{Matrix}\hfill \end{array}$$

(3)

$$\begin{array}{cccc}\hfill f\left(\delta u\right)& =\delta {\mathbf{u}}^T\mathbf{f}\hfill & \hfill & \Rightarrow \mathbf{f}:\mathrm{External}\mathrm{Structural}\mathrm{Force}\hfill \end{array}$$

(4)

$$\begin{array}{cccc}\hfill h(p,\delta p)& =\delta {\mathbf{p}}^T\mathbb{H}\mathbf{p}\hfill & \hfill & \Rightarrow \mathbb{H}:\mathrm{Acoustic}\mathrm{Stiffness}\mathrm{Matrix}\hfill \end{array}$$

(5)

$$\begin{array}{cccc}\hfill q(\delta p,p)& =\delta {\mathbf{p}}^T\mathbb{Q}\mathbf{p}\hfill & \hfill & \Rightarrow \mathbb{Q}:\mathrm{Acoustic}\mathrm{Mass}\mathrm{Matrix}\hfill \end{array}$$

(6)

In the equations above, k denotes the virtual structural potential energy, m the virtual structural kinetic energy, c the virtual coupling energy on the structural side, and f the virtual external work. By analogy between the structure and the fluid domains, h and q are introduced as the virtual acoustic potential energy and the virtual acoustic kinetic energy, respectively. Moreover, the matrices $\mathbb{K}$ and $\mathbb{H}$ are positive semi-definite; the kernel of $\mathbb{K}$ has dimension six in 3D, while the kernel of $\mathbb{H}$ has dimension one. The matrix $\mathbb{M}$ is positive definite. Based on Equations (1)–(6), the operator can be written in discretized form as:

$$\begin{array}{c}\hfill \left(\left[\begin{array}{ccc}\mathbb{K}& & -\mathbb{C}\\ \mathbf{0}& & \mathbb{H}\end{array}\right]-{\omega }^2\left[\begin{array}{ccc}\mathbb{M}& & \mathbf{0}\\ {\rho }_f{\mathbb{C}}^T& & \mathbb{Q}\end{array}\right]\right)\left[\begin{array}{c}\mathbf{u}\\ \mathbf{p}\end{array}\right]=\left[\begin{array}{c}\mathbf{f}\\ \mathbf{0}\end{array}\right]\end{array}$$

(7)

For further details, see [5,14].

2.2. 1D Immersed IGA

This section outlines the immersed IGA methodology. The objective is to demonstrate how a geometry can be reconstructed starting from its boundary definition only. CAD files generally provide a boundary representation of volumes, whereas numerical analyses, such as acoustic modal analyses in vibroacoustic problems, require a full volumetric description.

For clarity, the methodology is first presented in one dimension. The extension to 3D is straightforward, as it corresponds to a direct generalization of the 1D case.

A one-dimensional bar of length L is considered. The bar is discretized into $n_{elem}^0$ elements at level 0 (initial arbitrary mesh), and the polynomial degree r is also defined. The element length is

$$L_{elem}^0={\displaystyle \frac{L}{n_{elem}^0}}$$

(8)

and the corresponding number of basis functions is

$$n_{basis}^0=r+n_{elem}^0$$

(9)

At this stage, the displacement field $u^0\left(\xi \right)$ and the geometry $x^0\left(\xi \right)$ are approximated as follows:

$$\begin{array}{ccc}\hfill u^0\left(\xi \right)& =\sum _{i=1}^{n_{basis}^0}{u}_i{N}^0\ast i,r\left(\xi \right)x^0\left(\xi \right)\hfill & \hfill =\sum \ast {i=1}^{n_{basis}^0}{x}_i{N}_{i,r}^0\left(\xi \right)\end{array}$$

(10)

Consider the case in which the bar is cut at a point $x_{stop}\left({\xi }_{stop}\right)$, assumed to be located arbitrarily such that $x_{stop}\in [0,L]$ and ${\xi }_{stop}\in [0,1]$. Figure 1 illustrates the bar, the corresponding basis functions, and the position of $x_{stop}$.

The procedure consists of identifying the following:

• The element containing ${\xi }_{stop}$, denoted $e_{stop}$;
• The boundary effects (depending on r);
• The number of non-zero basis functions $n_{remove}^0$ in that element;
• The remaining basis functions $n_{keep}^0$.

Once the basis functions to be replaced at level 0 are identified, they are substituted with a new set of functions obtained through h-refinement. The refinement is performed such that the number of elements at level 1, denoted $n_{elem}^1$, is twice that of level 0:

$$n_{elem}^1=2\times n_{elem}^0\mathrm{and}{L}_{elem}^1={\displaystyle \frac{L}{n_{elem}^1}}$$

(11)

The replaced basis functions from level 0, together with the new basis functions introduced at level 1, are illustrated in Figure 2.

The field discretization then becomes

$$\begin{array}{cc}\hfill u^1\left(\xi \right)& =\sum _{i=1}^{n_{keep}^0}{u}_i{N}_{i,r}^0\left(\xi \right)+\sum _{i=1}^{n_{added}^1}{u}_i{N}_{i,r}^1\left(\xi \right)\hfill \end{array}$$

(12)

This procedure can be repeated for l levels and it is called the Hierarchical B-Spline (HBs) approach.

A remark should be made regarding the edge effects. This aspect is discussed in Appendix A.

Afterward, it is necessary to identify the Gauss points that lie inside the domain of interest and those that fall outside of it. In this case, since the objective is to cut the bar at the red point, all Gauss points located to the right of this point are penalized by applying a weighting factor, ensuring that they do not influence the domain of interest. Furthermore, the degrees of freedom associated with the excluded portion of the domain are removed from the analysis.

The approach discussed in the one-dimensional case can be extended to three dimensions by applying the same procedure along the three spatial directions.

3. Results

This section presents the analysis of the results for two application examples.

The first example consists of a 1D bar of length L, which is cut at a random point $x_{stop}$, after which a modal analysis is performed. The results are compared with those obtained from an exact solution available for this case.

The second example deals with a 3D vibroacoustic problem. The acoustic cavity has a parallelepiped shape, and only its boundary is known. First, the 3D acoustic cavity is reconstructed using an immersed IGA approach, and an acoustic modal analysis is carried out. The results are then compared with those of an exact solution. Finally, the reconstructed acoustic cavity is coupled with a Kirchhoff–Love plate to evaluate the coupled vibroacoustic behaviour.

3.1. 1D Example

In this section, a tensile–compressive bar of length $L=1$ is analyzed. The bar is cut at the point $x_{stop}=0.7231$, after which a modal analysis is performed on the segmented bar. The obtained results are then compared with those from an exact solution. Figure 3 illustrates the problem under consideration.

For the sake of simplicity, material parameters such as Young’s modulus and density are set to unity. The adopted parameter values are summarized in

Table 1. Structural and geometrical parameter of the bar.

E [Pa]	$\mathit{\rho }$ [$\frac{\mathbf{kg}}{{\mathbf{m}}^{\mathbf{3}}}$]	L [m]	${\mathit{x}}_{\mathit{stop}}$ [m]	r [-]	${\mathit{n}}_{\mathit{elem}}^0$
1.0	1.0	1.0	0.7231	6	8

.

Considering six levels (the initial level plus five refinement levels), the global basis functions considered are shown in Figure 4.

Figure 5 shows all the Gauss points of the considered problem. The Gauss points located outside the domain are penalized, while the integration is performed only on those lying inside the domain.

Figure 6 shows the first two numerical and analytical mode shapes, while

Table 2. First five structural numerical and analytical frequencies and their corresponding percentage error.

${\mathit{f}}_{\mathit{an}}$ (Hz)	${\mathit{f}}_{\mathit{num}}$ (Hz)	Error $(\%)$
0.346	0.346	0.03
1.037	1.037	0.03
1.729	1.729	0.03
2.420	2.420	0.03
3.112	3.112	0.03

reports the computed frequency values along with the corresponding error respect to the exact solution.

3.2. 3D Example

This section is devoted to a three-dimensional example. Consider a geometry obtained from CAD software, as illustrated in Figure 7, which provides only the boundary representation of the domain. The dimensions of the geometry are $1\times 1\times 0.111111$. For this example, all parameters (Young’s modulus, density, and sound speed) are set to 1, and three HB levels are constructed. The procedure to be followed consists of the following steps:

• Immerse the geometry into an 3D IGA computational domain;
• Determine the number of refinement levels required to achieve a suitable approximation of the involved operators;
• Identify the basis functions to be retained at each level;
• Select only the Gauss integration points located inside the domain of interest, which will be used to perform the numerical integration.

The workflow of the process described above is illustrated in Figure 8.

Figure 9 shows the distribution of the Gauss points, highlighting those inside and outside the domain of interest. The numerical integration is carried out only on the internal points, while the external ones are penalized.

Table 3. First acoustic numerical and analytical frequencies, as well as the corresponding percentage error.

${\mathit{f}}_{\mathit{an}}$ (Hz)	${\mathit{f}}_{\mathit{num}}$ (Hz)	Error (%)
0.500	0.500	0.001
0.707	0.707	0.001
1.000	1.000	0.002
1.118	1.118	0.010
1.414	1.414	0.011

reports the results of the first five numerical and analytical frequencies together with the corresponding percentage error. Furthermore, Figure 10 illustrates the first two numerical mode shapes.

3D Coupling

The coupling is performed at the 2D NURBS interface surface. The Gauss points in the two-dimensional structural parametric space are considered as integration points. In short, the integration over a 2D IGA surface involves two transformations. The main goal is to construct the coupling matrix:

$$\begin{array}{cc}\hfill c(p,\delta \mathbf{u})={\int }_{{\Sigma }_i}p\mathit{n}·\delta \mathbf{u}dS& =\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & ={\int }_{{\widehat{\Sigma }}_i}{p}^h\left({\mathit{\xi }}_f\left({\mathit{\xi }}_s\right)\right)\mathit{n}·\delta {\mathbf{u}}^h\left({\mathit{\xi }}_s\right)J_1\left({\mathit{\xi }}_g\right)J_2\left({\mathit{\xi }}_s\right)d\widehat{S}\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & \simeq \sum _{g=1}^{n_G}{p}^h\left({\mathit{\xi }}_f\left({\mathit{\xi }}_g\right)\right)\mathit{n}·\delta {\mathbf{u}}^h\left({\mathit{\xi }}_g\right)J_1\left({\mathit{\xi }}_g\right)J_2\left({\mathit{\xi }}_g\right)w_g\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \simeq \delta {\mathbf{u}}^T\left[\sum _{g=1}^{n_G}{\mathbf{R}}^T\left({\mathit{\xi }}_g\right)\mathbf{n}\mathbf{S}\left({\mathit{\xi }}_f\left({\mathit{\xi }}_g\right)\right)J_1\left({\mathit{\xi }}_g\right)J_2\left({\mathit{\xi }}_g\right)w_g\right]\mathbf{p}\hfill \end{array}$$

(16)

At these points, it is necessary to evaluate the values of the NURBS basis functions on the structural side ($\mathbf{R}\left({\mathit{\xi }}_g\right)$), as well as the values of the HB functions on the fluid side (arising from the immersed IGA) ($\mathbf{S}\left({\mathit{\xi }}_f\left({\mathit{\xi }}_g\right)\right)$). In the previous formula, $J_1$ and $J_2$ are the Jacobian of the two transformations.

To verify whether the coupling matrix has been properly constructed, a quasi-static analysis is performed on the problem illustrated in Figure 11. A displacement field equal to the structural mode shape itself is applied to the acoustic cavity. This procedure is repeated for the first four structural vibration modes, and a qualitative assessment is made to determine whether the resulting pressure field reproduces the displacement field.

The governing equation applied is

$$\begin{array}{c}\hfill \mathbf{P}={\mathbb{H}}^{+}\mathbb{C}{\Phi }_{\mathbf{s}}\end{array}$$

(17)

where $\mathbf{P}$ denotes the pressure field, ${\mathbb{H}}^{+}$ is the pseudo-inverse of the acoustic stiffness matrix, $\mathbb{C}$ is the coupling matrix, and ${\Phi }_s$ represents the structural modal basis.

Figure 12 shows the first four structural modes and the corresponding acoustic response induced by these modes. The obtained pressure field is completely consistent with the structural modes.

4. Discussion and Conclusions

The results presented in this conference proceeding are very promising for future full integration between IGA and CAD. The outcomes obtained in both 1D and 3D cases are highly accurate with respect to exact solutions and can be readily extended to more complex geometries.

Future developments should focus on integrating the immersed IGA approach with complex geometries composed of multiple patches. Other possible research directions include investigating whether truncated B-splines offer actual advantages compared to hierarchical ones.