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Certifying performance of mechanical components with experimental tests is time consuming and expensive, which motivates the development of efficient approaches for predicting the outcomes from such testing. We propose two methods based on Gaussian processes (GP) to estimate the probability that new components will pass future certification tests, while assessing our prediction confidence. The first method processes a set of Bernoulli trials into a suitable machine learning dataset and subsequently infers the probability of performing satisfactorily for new components using heteroscedastic bounded GP regression. The second method uses GP classification with linear kernels. We demonstrate that linear kernels are well suited for datasets representing snapshots of mechanical system responses by accurately reproducing the underlying physical trends in the data. This yields consistent probabilities of passing and provides high labeling accuracy, even with small datasets. We demonstrate these techniques on synthetic datasets consistent with ship cabinet certification tests. We achieve up to 100% accuracy using all of the training data, and at least 92% with only 10% of the available data. With a corrupted training set, we obtain at least 93% accuracy. In the regression framework, we demonstrate that introducing heteroscedasticity helps achieve significantly better accuracy than frequentist machine learning methods.

Despite the enormous strides in computing power, reduced-order models (ROMs) remain essential for design optimization and early stage design. The energetically optimal basis for ROMs of vibroacoustic systems derives from the coupled Eigen problem: no basis of smaller dimension contains a greater amount of system energy. While vital, this L2-type framework is also insufficiently accurate on a pointwise basis for many problems. A simply supported beam with a point load is an obvious thought experiment: the smooth modal functions converge to the correct displaced shape very slowly. The well-known “residual mode” or orthocomplement concept overcomes this problem, by augmenting the space of modal functions with static solutions to the point loads. These new residual modes are orthogonalized against the modal space, and used otherwise conventionally. This work describes the development of ROMs of submerged structures by extending the residual mode approach to the strongly coupled structural-acoustic case. We derive methods to use these ROMs in substructures. Numerical examples show that the structural acoustic residual mode formulation is very effective in improving the solution method in the low-frequency limit, especially in the fluid, and also at the high-frequency limit, where response quantities tend to the average of the direct-solve FEM response.Despite the enormous strides in computing power, reduced-order models (ROMs) remain essential for design optimization and early stage design. The energetically optimal basis for ROMs of vibroacoustic systems derives from the coupled Eigen problem: no basis of smaller dimension contains a greater amount of system energy. While vital, this L2-type framework is also insufficiently accurate on a pointwise basis for many problems. A simply supported beam with a point load is an obvious thought experiment: the smooth modal functions converge to the correct displaced shape very slowly. The well-known “residual mode” or orthocomplement concept overcomes this problem, by augmenting the space of modal functions with static solutions to the point loads. These new residual modes are orthogonalized against the modal space, and used otherwise conventionally. This work describes the development of ROMs of submerged structures by extending the residual mode approach to the strongly coupled structural-acoustic case. We de...

The design of broadband mechanical metamaterials in the context of acoustics can be driven by the invariance of the wave equation under a set of special coordinate transformations called transformation acoustics. A program of homogenization to design the metamaterial structure can be derived from these transformation functions subject to geometric constraints from the metamaterial’s intended application. The limits of homogenization in a nonperiodic, nonlinear environment however must be accounted for and corrected. This can be done by means of nonlinear manifold interpolation methods, the feedback into which is driven by the scattered wave field in the ambient medium. The feedback is minimized using statistical and machine learning techniques, dictating the final metamaterial structure. The performance of this algorithmic method is compared against a “brute force” optimization approach driven by a convolutional neural network with no transformation of the underlying wave equation.The design of broadband mechanical metamaterials in the context of acoustics can be driven by the invariance of the wave equation under a set of special coordinate transformations called transformation acoustics. A program of homogenization to design the metamaterial structure can be derived from these transformation functions subject to geometric constraints from the metamaterial’s intended application. The limits of homogenization in a nonperiodic, nonlinear environment however must be accounted for and corrected. This can be done by means of nonlinear manifold interpolation methods, the feedback into which is driven by the scattered wave field in the ambient medium. The feedback is minimized using statistical and machine learning techniques, dictating the final metamaterial structure. The performance of this algorithmic method is compared against a “brute force” optimization approach driven by a convolutional neural network with no transformation of the underlying wave equation.

Large-scale analysis of transducer arrays usually requires detailed, high-resolution modeling to represent the coupling effects of piezoelectric, elastic, and fluid media. However, reduced-order models, in the form of low-DOF lumped-parameter engineering models remain essential for system design. Here, we show how to bridge the gap between these model classes by deriving a simple means to compute the modes of an arbitrary piezoelectric system. Our approach exploits the intrinsic disparity in time scale between the electric and elastic fields to reduce the coupled problem into an augmented, elastic-only, system of equations. This system is shown to have the same mathematical characteristics as the elastic system, insofar as no new singularities appear. The piezoelectric-elastic modal system is fully general, and compatible with large-scale finite element representations of complex transducer designs. Numerical examples are shown to demonstrate the efficiency and practicality of the modal analysis.Large-scale analysis of transducer arrays usually requires detailed, high-resolution modeling to represent the coupling effects of piezoelectric, elastic, and fluid media. However, reduced-order models, in the form of low-DOF lumped-parameter engineering models remain essential for system design. Here, we show how to bridge the gap between these model classes by deriving a simple means to compute the modes of an arbitrary piezoelectric system. Our approach exploits the intrinsic disparity in time scale between the electric and elastic fields to reduce the coupled problem into an augmented, elastic-only, system of equations. This system is shown to have the same mathematical characteristics as the elastic system, insofar as no new singularities appear. The piezoelectric-elastic modal system is fully general, and compatible with large-scale finite element representations of complex transducer designs. Numerical examples are shown to demonstrate the efficiency and practicality of the modal analysis.

The acoustic cloaking theory of Norris (2008) permits considerable freedom in choosing the transformation function f from physical to virtual space. The standard process for defining cloak materials is to first define f and then evaluate whether the materials are practically realizable. In this paper, this process is inverted by defining desirable material properties and then deriving the appropriate transformations which guarantee the cloaking effect. Transformations are derived which result in acoustic cloaks with special properties such as 1) constant density 2) constant radial stiffness 3) constant tangential stiffness 4) power-law density 5) power-law radial stiffness 6) power-law tangential stiffness 7) minimal elastic anisotropy., 9 pages, 8 figures

Transformation acoustics uses invertible maps to transform a cloaked region to a region with a smaller scatterer. The “elastic” or Norris class of acoustic cloaking theory is inherently broadband, in that it does not rely on the presence of resonant effects. To achieve the properties required by the transformation theory, we utilize pentamode materials, which admit the use of finite mass but require anisotropic stiffness throughout the material. These types of materials do not exist naturally and must be fabricated through the use of metamaterials. One of the challenges associated with metamaterials for use in dynamic applications is the distribution of mass in a unit cell and its consequences for frequency-dependent behavior in the metamaterial. Various homogenization techniques for recovering wave speed properties of the metamaterial unit cells may predict contradictory wave speed values for pentamodal structures. These disagreements can be explained through the distribution of inertia in the material a...

Functionally graded acoustic metamaterials (FGAMs) can be designed to have specific waveguide properties dictated by a theory relevant to the application. Frequently, these material properties do not exist naturally, and must be fabricated by gradually layering manufactured unit cell microstructures, resulting in a (usually smooth) variation of properties. Tailoring these microstructures to the demands of the relevant theory requires assembling microstructures with very specific material properties—a process that often requires haphazardly searching a large domain space of unit cells for desirable parameter combinations. Moreover, the process of determining the material tensor of interest for a specific set of metamaterial cell design parameters typically involves solving a costly high dimensional finite element problem for each new microstructure cell of interest. This work presents a gradient-based, manifold interpolation technique to characterize acoustic metamaterial homogenized elastic tensors. An ad...

The field of acoustic metamaterial research has been driven, first, by transformation acoustics cloaking theory. This and other theoretical approaches optimize an acoustic effect through definition of artificial material properties which at present are only achievable using metamaterial technology. Metamaterials, however, present separate challenges in optimization for any desired acoustic effect: in particular, their dynamic behavior depends on parameters unrelated to acoustics, and exhibits wave propagation behavior more complex than elastic or acoustic continua. Homogenization methods which assume Cartesian symmetry are a staple in metamaterial design, but these only approximate a feasible optimal design. Moreover, total reliance on homogenized continuum models provides no information about the actual microstructure performance, and presents problems in functionally graded applications. To overcome this, we augment our Cartesian homogenization processes with high-resolution finite element models to optimize the design. Comparatively computationally expensive implicit FEM is avoided; specifically tailored time-domain wave propagation codes make the analyses feasible. Examples of the process, combining Cartesian homogenization estimates with high-resolution microstructural wave propagation solutions, will be shown.The field of acoustic metamaterial research has been driven, first, by transformation acoustics cloaking theory. This and other theoretical approaches optimize an acoustic effect through definition of artificial material properties which at present are only achievable using metamaterial technology. Metamaterials, however, present separate challenges in optimization for any desired acoustic effect: in particular, their dynamic behavior depends on parameters unrelated to acoustics, and exhibits wave propagation behavior more complex than elastic or acoustic continua. Homogenization methods which assume Cartesian symmetry are a staple in metamaterial design, but these only approximate a feasible optimal design. Moreover, total reliance on homogenized continuum models provides no information about the actual microstructure performance, and presents problems in functionally graded applications. To overcome this, we augment our Cartesian homogenization processes with high-resolution finite element models to opt...

A previously reported structural-acoustic frequency-domain formulation for layered, ribbed structures is used here as the basis for an approach to optimize these systems. We will review a previously reported singular perturbation approach to resolve convergence difficulties in both planar and cylindrical configurations, and discuss verification and validation. The significant successes of topological structural optimization for strength, weight, and efficiency are more challenging to repeat in coupled wave-bearing systems with many modes present. This is due, at least, to the difficulties of defining appropriate cost and regularization functions applicable across frequency ranges of interest and across the orders of magnitude of response characteristic of vibroacoustic systems. We discuss overcoming these difficulties by evading them entirely: the layered ribbed structural-acoustic model we use is sufficiently fast that sophisticated optimization algorithms are not required. To understand the dependence o...

Functionally graded acoustic metamaterials (FGAMs) can be designed to have specific waveguide properties. In sonar applications, FGAMs can be tailored to resist incident wave reflection. As these materials do not exist naturally, they must be fabricated by gradually layering manufactured, resulting in a (usually smooth) variation of properties. Validating material properties of FGAMs is difficult with conventional tests, as the distribution of material properties over the layered structure results in a non-unique solution if typical data (e.g., compressive strain) is measured. This talk will demonstrate an approach to characterize the functionally graded material properties by parameterizing how the functionally graded material changes throughout the specimen. Experiments designed to minimize the uncertainty surrounding the FGM model parameters are formulated and evaluated numerically. The FGM model parameters are estimated by Markov chain Monte Carlo so that a probability distribution surrounding each parameter is recovered. Probability distributions enable uncertainty quantification (UQ) surround the validated material parameters. UQ is important as uncertainty surrounding the parameter results directly translates into uncertainty surround the FGM performance. The approach is verified by bootstrap analyses of known FGAM distributions.

This study examines the theory and performance of Jacobian-Free Newton Krylov (JFNK) methods for the efficient, iterative solution of steady state dynamics of linear vibrating systems in the frequency domain. Currently, most commercial FEM algorithms employ use direct factorization of large system matrices to achieve steady state solution. Such approaches are usually quite demanding on memory and CPU requirements. Some implementations exploit iterative solutions, but still use large, assembled system matrices, requiring significant memory. The methods investigated in this work avoid the formation of a matrix completely, minimizing memory requirements and enabling much larger problems to be performed on desktop computers. Here, the Conjugate-Gradient (CG) and Transpose Free Quasi Minimal Residual (TFQMR) algorithms are studied as possibilities. These methods are of particular interest for adaptation to finite element software which uses explicit transient dynamics, because such software's optimal architecture prevents the formation and solution of a stiffness (or a tangent stiffness) matrix. In order to demonstrate the advantages of these algorithms, we choose examples that use the JFNK-CG and JFNK-TFQMR technique to show computational advantages over regular matrix based solutions, with significant decrease in memory requirements.

Fluid filled foams are important components of acoustical systems. Most are made up of a stiff skeleton medium relative to the fluid considered, usually air. Biot’s theory of poroelasticity is appropriate for characterizing and modeling these foams. The use of relatively stiff fluid (such as water) and limp foam media pose a greater challenge. Recently modifications to Biot's theory have generated the mechanical relationships required to model these systems. Necessary static material properties for the model can be obtain through in vacuo measurement. Frequency dependent properties are more difficult to obtain. Traditional impedance tube methods suffer from fluid structure interaction when the bulk modulus of the fluid media approaches that of the waveguide. The current investigation derives the theory for, and investigates the feasibility of, several rigid impedance tube alternatives for characterizing limp foams in stiff fluid media. Alternatives considered include a sufficiently rigid impedance tube, a pressure relief impedance tube and, the most promising, a piston excited oscillating chamber of small aspect ratio. The chamber concept can recover the descriptive properties of a porous medium described by Biot’s theory or by complex-impedance equivalent-fluid models. The advantages of this facility are small facility size, low cost, and small sample size.

Noise in an acoustic array can be defined as any unwanted signal, and understanding how noise interacts with a structural system is paramount for optimal design. For example, in an underwater vehicle we may want to understand how structural vibrations radiate through a surrounding fluid; or an engineer may want to evaluate the level of sound inside a car resulting from the turbulent boundary layer (TBL) induced by a moving vehicle. This talk will discuss a means of modeling noise at a point of interest (e.g., at a sensor location) stemming from a known source by utilizing a power transfer function between the source and the point of interest, a generalization of the work presented in [1]. The power transfer function can be readily computed from the acoustic response to an incident wave field, requiring virtually no additional computation. The acoustic solution may be determined via analytic frequency domain approaches or through a finite element analysis, enabling the noise solution to be a fast post processing exercise. This method is demonstrated by modeling the effects of a TBL pressure and noise induced by structural vibrations on a sensor array embedded in an elastic, multi-layer solid. Additionally, uncertainties in the noise model can be easily quantified through Monte Carlo techniques due to the fast evaluation of the noise spectrum. Ko, S.H. and Schloemer, H.H. “Flow noise reduction techniques for a planar array of hydrophones,” J. Acoust. Soc. Am. 92, 3409 (1992).

An existing analytical, frequency domain solution for wave propagation in coated, ribbed, three-dimensional elastic layered plates excited by acoustic plane waves provides fast solutions for high frequency excitations. The solution methodology, which is found to be numerically unstable under certain conditions, contains an Ansatz for a particular wave number expansion in the direction of periodicity. Evidence is presented to show that the numerical instability is due to the specific choice of the wave number basis. In order to provide a remedy while retaining the positive aspects of the solution methodology, we determine the set of admissible propagating (and attenuating) waves via an eigenvalue analysis. Several approaches exist to determine the admissible waves of structures with periodicity. The Wave Guide Finite Element (WFE) method leads to a two parameter, nonlinear eigenvalue problem, which is difficult to solve. The Scale Independent Element (SIE) formulation results in a two-parameter quadratic eigenvalue problem and overcomes the numerical issues of the WFE method. This study examines available methods in determining the admissible waves and compares them with those established using the dispersion relation. The computed admissible waves are then compared with the aforementioned Ansatz.

An existing pseudo-analytical, frequency domain solution for wave propagation in coated, ribbed, three-dimensional elastic layered plates excited by acoustic plane waves provides fast solutions for high frequency excitations. The solution methodology, which is found to be numerically unstable under certain conditions, contains a fundamental ansatz regarding the set of excited wave forms expressed through a particular wave number expansion in the direction of periodicity. We propose to identify the set of admissible propagating (and attenuating) waves via an eigenvalue analysis as a preprocessing step when executing the pseudo-analytical solution. A significant challenge lies in determining the admissible waves of structures with periodicity in two-dimensions. The Wave Guide Finite Element (WFE) method leads to a two parameter, nonlinear eigenvalue problem, which is extremely difficult to solve. By formulating the problem with the Scale Independent Element (SIE) formulation, the admissible waves can be expressed through a two-parameter quadratic eigenvalue problem. This formulation overcomes the numerical conditioning issues associated with WFE method. This study compares the aforementioned ansatz with that computed by the SIE formulation as well as the improvements in the numerical stability associated with this additional step. The forced response results are compared to a finite element analysis.

There is a need for a fast and reliable tool to assist in the analysis, design, and optimization of submarine and UUV coatings due to high frequency incident acoustic pressure loading. An existing pseudo-analytical, frequency domain solution for wave propagation in coated, ribbed, three-dimensional elastic layered plates excited by acoustic plane waves provides fast solutions for high frequency excitations. Weidlinger Associates, Inc. (WAI) is developing an analysis software tool which integrates this solution methodology while adding some technical improvements to the formulation. The solution methodology, which is found to be numerically unstable under certain conditions, contains a fundamental ansatz regarding the set of excited wave forms expressed through a particular wave number expansion in the direction of periodicity. Evidence is presented to show that the numerical instability is due to the specific choice of the wave number basis used in the solution. In order to provide a remedy while retaining the positive aspects of the solution methodology, WAI is implementing a pre-processing step to determine the optimal wave number basis: the set of admissible propagating (and attenuating) waves are predetermined via an eigenvalue analysis and then substituted into the wave number basis in computing the pseudo-analytical solution.

Acoustic metamaterials use sub-wavelength, anisotropic, and inhomogeneous microstructures. Macroscopic properties can be related to the microstructure using homogenization theory Hassani and Hinton [Comput. Struct. 69, 719–738 (1998), which allows an analyst to confirm the extent to which a candidate metamaterial microstructure meets the requirements for a pentamode cloaking material. Norris [Proc. R. Soc. Ser. A, 464, 2411–2434 (2008)] presented a theory of transformation acoustics that enables the realization of inhomogeneous pentamode acoustic materials having anisotropic elastic tensors, isotropic density and finite mass. This theory describes the spatially varying material properties in terms of a mapping, which for separable geometries may be generated using a scalar function. This function, the constraints on its behavior implicit in the Norris theory, and the material equations constitute the defining relations for pentamode transformation acoustics. Previously, analytic work in transformation acoustics developed the material properties after having fixed a transformation. By reversing the process, we create a number of new families of pentamode cloaking materials. We validate the concept with three-dimensional explicit transient finite element simulations.

Norris’ theory of transformation acoustics enables the realization of pentamode acoustic materials having anisotropic density and finite mass. Two additional extensions of the theory are considered, with a view toward demonstrating feasibility for practical applications. First, specializations of the theory developed by Norris [“Acoustic cloaking theory,” Proc. R. Soc. London, Ser. A 464, 2411–2434 (2008)] are considered in order to develop transformations corresponding to pentamode acoustic metamaterials with specified functional forms (constant or powerlaw) for density and stiffness. Ray‐tracing and finite element simulations of wave propagation through such pentamode acoustic media are presented. Next, the design of acoustic materials for objects composed of simple geometric shapes (e.g., a cylinder with spherical endcaps) is considered. It is shown that certain classes of transformations are well‐suited for the design of such shapes and validate the concept with finite element simulations.

Norris [“Acoustic cloaking theory,” Proc. R. Soc. London, Ser. A 464, 2411–2434 (2008)] presented a theory of transformation acoustics that enables the realization of pentamode acoustic materials having anisotropic density and finite mass. The design of acoustic materials for 3‐D objects composed of combinations of simple geometric shapes, e.g., a cylinder with spherical end‐caps, is considered. Certain classes of transformations which are well‐suited for the design of such shapes are described and conditions under which these transformations result in materials which minimize acoustic reflections are proposed. The concept is validated using 3‐D explicit transient finite element simulations, which closely emulate the physics of wave propagation in the proposed materials.

Acoustic metamaterials need to be realized using subwavelength microstructures. By tailoring the microstructure of the underlying unit cell, different effective properties at the macroscopic scale may be achieved. These macroscopic properties can be related to the microstructure using homogenization theory. The problem of designing a pentamode acoustic metamaterial is formulated as the minimization of an objective functional representing the difference between the homogenized properties and the target pentamode acoustic metamaterial properties. A quasi‐Newton optimization approach is used to solve the optimization problem. Such an approach requires the gradient of the objective function with respect to the microstructural properties at every iteration, and hence the cost of gradient computation must be low. The gradient is calculated efficiently using the adjoint approach which requires the solution of only two (primal and dual) finite element problems, per homogenization deformation field, independent of...

Determination of characteristic frequencies and solutions (mode shapes) for vibrating systems is a staple of linear dynamics in engineering, so it should be no surprise that analysts strive for the same in coupled structural‐acoustic systems. Computation of these eigensolutions confers not only physical insight into the behavior of the system, but also a means to reduce the mathematical dimension of frequency‐response and other engineering computations. As in other mechanical vibration problems, the eigensolutions of coupled systems form a nearly optimal space for the approximation of the forced‐response solutions. However, the coupled structural‐acoustic eigenproblem is not trivial, and can be prohibitively expensive if not handled properly; this is particularly true for exterior problems. Notably, identification of a physically pertinent self‐adjoint operator for the system of interest is a critical challenge. Here we will discuss the coupled structural‐acoustic eigenproblem, various approximations comm...

Reduced order models are developed for fully coupled structural‐acoustic unsymmetric matrix models, resulting from Cragg’s displacement/pressure formulation, using Krylov subspace techniques. The reduced order model is obtained by applying a Galerkin and Petrov‐Galerkin projection of the coupled system matrices, from a higher dimensional subspace to a lower dimensional subspace, while matching the moments of the coupled higher dimensional system. Two such techniques, based on the Arnoldi algorithm, focusing on one‐sided and two‐sided moment matching, are presented. To validate the numerical techniques, an ABAQUS coupled structural‐acoustic Benchmark problem is chosen and solved using the direct approach. First, the physical problem is modeled using ANSYS FE package and compared with closed form solutions. Next, ANSYS results are compared with nodal velocities obtained by generating reduced order models via moment matching. The results show that the reduced order models give a very significant reduction in...

Infinite elements are distinct from alternative methods for exterior problems in that, like a finite element, they describe a small sub‐region of the problem domain, and use locally supported shape functions to derive a method of weighted residuals statement thereon. Until recently, transient analyses using explicit time‐integration methods, and eigenanalysis, were not possible using infinite elements. A formulation of acoustic infinite element compatible with implicit dynamics, time‐harmonic acoustics, far‐field extrapolation, explicit dynamics, and eigenanalysis is described. First steps in the development are the adoption of a basis corresponding to spherical radiating harmonics, the Bettess geometric map, and the Astley‐Leis weighted residual formulation. A modified means to compute the element integrals, and a modification of the spherical harmonic basis, improve numerical conditioning of the element and stability of the formulation. The trivial frequency dependence of the Astley‐Leis formulation, cr...

Numerical studies of various infinite element formulations are conducted: conjugated types, unconjugated, decay (unconjugated), and DAA1 elements of the Olsen/Bathe type. In addition, an ‘‘element’’ implementing the Bayliss et al. first‐order condition is studied. All elements except the decay are implemented in separable coordinates. The element formulations are studied analytically, using axisymmetric modal impedances of a sphere. These studies reveal limitations, advantages, and appropriate methods of usage for the various formulations. Subsequently, the various elements are implemented in production‐type software, and numerical comparisons are conducted. These studies reveal the relative accuracy of the formulations as well, but also permit discussion of numerical performance. Fully coupled, three‐dimensional structural acoustic meshes are studied. Comparisons are made against a BEM method, and analytic solutions where appropriate. When used properly, infinite element methods can deliver accuracy comp...