Analytical/numerical Matching Research Articles

Acoustic reflection and transmission by sub-critical and super-critical elastic partitions with structural discontinuities forced by multiple-angle oblique broadband sound waves

An analysis is presented for acoustic reflection and transmission from an infinite fluid-loaded flexible barrier with spatially periodic discontinuities. The plate, with similar or dissimilar acoustic fluids on both sides, is excited by an oblique wave incident acoustic field. The fully-coupled structural/acoustic problem is treated by Analytical-Numerical Matching (ANM) which improves the numerical accuracy and convergence rate. Periodic spatial discontinuities, modelled by various boundary conditions, create deviations from specular directivity. For super-critical structural wave speeds, the scattering is characterized by a distribution of reflection and transmission angles centered around the Mach Angles. For sub-critical structural waves, energy redistribution is most pronounced at the structural resonances. These results are compared to a baffled finite barrier analyzed by approximate means. For the subcritical finite barrier, a fluid loading correction is introduced to the structural wavenumber to approximate the effects of the fluid coupling and radiation damping. For subsonic flexural waves, the radiation is analyzed in terms of the wavenumber transform and interpreted as a series of edge singularities dependent on the boundary conditions. For the super-critical configuration, the effects of the fluid loading are introduced by approximating the structural wavenumber from the dispersion relation of the free response of the unbounded structure.An analysis is presented for acoustic reflection and transmission from an infinite fluid-loaded flexible barrier with spatially periodic discontinuities. The plate, with similar or dissimilar acoustic fluids on both sides, is excited by an oblique wave incident acoustic field. The fully-coupled structural/acoustic problem is treated by Analytical-Numerical Matching (ANM) which improves the numerical accuracy and convergence rate. Periodic spatial discontinuities, modelled by various boundary conditions, create deviations from specular directivity. For super-critical structural wave speeds, the scattering is characterized by a distribution of reflection and transmission angles centered around the Mach Angles. For sub-critical structural waves, energy redistribution is most pronounced at the structural resonances. These results are compared to a baffled finite barrier analyzed by approximate means. For the subcritical finite barrier, a fluid loading correction is introduced to the structural wavenumber to a...

Reflection and transmission of acoustic energy by an elastic plate with structural discontinuities forced by multiple angle oblique broadband sound waves

An analysis is presented for acoustic reflection and transmission from an infinite fluid-loaded plate with spatially periodic discontinuities. The plate, with similar or dissimilar acoustic fluids on both sides, is excited by an oblique wave incident acoustic field. The fully-coupled structural/acoustic problem is treated by the method of Analytical-Numerical Matching (ANM). The ANM framework separates the problem into global numerical and local analytical solutions, and handles rapid spatial variation around the structural discontinuities in closed form, improving the numerical accuracy and convergence rate. The ANM approach includes a novel way to handle difficulties associated with coincidence frequencies. The periodic spatial discontinuities, modeled by various boundary conditions, create deviations from specular directivity with multiple reflection and transmission angles, the effect being most pronounced at structural resonances. The periodic discontinuities redirect part of the structural energy in...

Prediction of reflection and transmission by an elastic barrier with periodic structural discontinuities forced by oblique acoustic waves

An analysis method is developed for acoustic reflection and transmission from an infinite fluid-loaded flexible barrier with spatially periodic discontinuities. The barrier has acoustic fluids on both sides, which can be dissimilar. The structure is excited by an oblique-wave incident acoustic field. The fully-coupled structural/acoustic problem is treated by the method of Analytical-Numerical Matching (ANM). The ANM framework separates the problem into a global numerical solution and local analytical solutions. ANM handles rapid spatial around the structural discontinuities, improving the accuracy and convergence rate of reflected and transmitted pressures. Furthermore, the ANM approach offers a way to handle the mathematical difficulties associated with coincidence frequencies. The periodic spatial discontinuities create variations from simple specular-like directivity with multiple reflection and transmission angles, the effect being most pronounced at structural resonances. The periodic discontinuities can be thought of as redirecting a portion of the structural energy into resonant substructures having wavenumbers different from the oblique wave forcing, reradiating reflection and transmission fields with different directivity patterns. Discrete frequency and broadband results are presented. The goal is to develop efficient first-principles methods for structural-acoustic reflection and transmission between coupled acoustic spaces and into surrounding media.

Prediction of broadband high-frequency acoustic reflection and transmission from elastic plates with structural discontinuities

A model for the acoustic reflection and transmission for flexible boundaries with structural discontinuities is developed. The system studied is an infinite, fluid loaded plate with periodically fixed constraints that is driven by an incident acoustic field. To treat the local discontinuities of the plate, the coupled structural/acoustic problem is treated by applying the method of Analytical-Numerical Matching (ANM). The ANM framework, which separates the problem into a global numerical solution and a local analytical solution, offers an approach to handle mathematical difficulties associated with the coincidence frequencies of the system. Accurately treating the rapid spatial variation around the discontinuities improves the overall accuracy and convergence of the computational analysis to predict the resulting pressure levels. Results for an infinite plate with discontinuities are compared to a baffled plate with a fluid loading correction introduced to the structural wavenumber. Simple closed-form ban...

Analytical-numerical matching for fluid-loaded structures with discontinuities

The method of analytical-numerical matching (ANM) is applied to several example problems involving the radiation and vibration of a fluid-loaded cylindrical shell with structural discontinuities. These problems are used to verify the accuracy of the method compared to a purely numerical approach, and to demonstrate the usefulness of ANM for solving problems involving nonaxisymmetric forcing and constraints on structures that are otherwise axisymmetric. The principal advantage of the ANM method for this application is its ability to employ a spectral method (modal decomposition in azimuth) in the solution of a problem that is nonaxisymmetric. ANM divides the original problem into local and global subproblems that are solved separately. The discontinuity is handled using a high-resolution local finite-element model. Smooth forces are derived from solving an analytical matching problem on the local domain, and these smooth forces are then applied to a low-resolution numerical global problem. The superposition of solutions to the subproblems is equivalent to the solution of the original problem. For fluid loading, the local high-resolution model is solved in vacuo, and the effects of fluid loading are included completely by the low-resolution global problem.

A viscous-acoustic boundary element formulation based on analytical-numerical matching

Analytical-numerical matching (ANM) is an analysis scheme that combines a low-resolution global numerical solution with a high-resolution local analytical solution and a matching solution to form a uniformly valid composite solution. The application of ANM to harmonically oscillating bodies in a fluid leads to a novel reformulation of boundary element problems in acoustics and aeroacoustics. The singular kernal encountered in the integral equation is handled simply within the analytical local solution. The numerical implementation utilizes a smoothed Green’s function solution to the governing equation with a distributed source term. Because the singular behavior has been removed from the numerical aspect of the problem, the approach converges rapidly and exhibits insensitivity to node (control point) location. The method allows low-resolution numerics to be combined with analytical corrections to obtain high accuracy solutions using a robust calculation scheme. The method has recently been extended to include viscous effects within the local solution. The appropriately modified global solution remains irrotational and can be expressed in terms of a smoothed potential. Sample results are shown for radiation from plates up to high frequencies. Ongoing research on ANM BEM is described, including work to include nonlinear convection within the viscous flow effects.

New analysis methods for the calculation of interior noise

Recently developed analysis methods applicable to the prediction of aircraft interior noise are described. Analytical-numerical matching (ANM) and Local-global homogenization (LGH) are methods for the low- to mid-frequency range. Energy-intensity boundary elements are applicable for high frequencies. ANM is a method that replaces the effect of structural discontinuities (ribs, stringers, attachments) with dynamically equivalent smooth force distributions, thereby removing the need for high-resolution numerics in the overall computational problem. LGH, which applies for periodic and quasiperiodic structures, provides a means to solve directly for the long wavelength spatial content of the structure. This part of the structural motion is strongly coupled most to the interior sound field. Energy-intensity boundary elements provide a very efficient means to predict high-frequency broadband sound fields in a fundamentally correct manner. Current research on this boundary element method involves its extension to handle coupling with structural surfaces vibrating at high frequency. The strengths and limitations of each of the above methods is discussed in the context of the interior noise problem. Innovative methods for the reduction of interior noise, as suggested by these approaches, are also mentioned.

Prediction of Aerodynamic Interactions of Helicopter Rotor on its Fuselage

An iterative and full-coupled rotor/fuselage aerodynamic interaction analytical method is developed based upon the rotor free wake model and the 3-D fuselage panel model. A close vortex/surface interaction model using the Analytical/Numerical Matching (ANM) was adopted in the method in order to simulate effectively the unsteady close interaction between the rotor tip-vortex and fuselage surface. By the analytical method, the unsteady and steady pressure distribution on the fuselage surface, and the unsteady lift and pitching moment of the fuselage in a rotor interaction environment were calculated for different advance ratios. It is shown that the unsteady aerodynamic loads of the fuselage due to the rotor interaction have the same periodic characteristics as the rotor. The comparisons between the present close vortex/surface interaction model and a previous model, which simply excludes vortex filaments inside the fuselage, were also made and the advantages of the former over the latter were demonstrated in improving unsteady close interaction calculations.

Local/global methods for structural acoustic radiation and scattering

Solving structural acoustic problems in the mid-frequency range is a challenging task. Computation is expensive due to the large well-coupled nature of the fluid-structure system. Discrete structural elements, e.g., ribs and stiffeners, cause localized discontinuities that require much greater resolution than would be otherwise warranted. Radiation and scattering problems having certain types of constraints exhibit great sensitivity to fine-scale convergence. Our purpose in this research is to devise analytical and numerical ways to recast such problems to be amenable to an efficient and accurate numerical solution. Analytical/Numerical Matching (ANM) decomposes a problem into local and global parts. Analysis methods extract the local discontinuities, replacing the original discrete influence of the inhomogeneity by smooth forces. A smoother computational problem can be solved efficiently, without loss of information. The method has been demonstrated for configurations of increasing complexity, and has become a useful tool for solving realistic problems. The ANM approach has been successfully applied to the modal/FEA structural-acoustics code sonax, providing a new capability to handle 3-D structural discontinuities. Future research on the application of ANM to other finite element structural-acoustic codes is outlined. An ongoing effort to develop a general theory of structural acoustic homogenization is also summarized.

Improvement of the solution convergence rate for an elastic shell driven by a vibratory constraint using analytical-numerical matching

A three-dimensional elastic shell is driven by the motion of an attached support and the solution is constructed using the method of analytical-numerical matching (ANM) with finite-element analysis and the structural-acoustic scattering code sonax. First, the area around the structural constraint is resolved in great detail as a separate local three-dimensional FEA solid model. Second, a Love–Timoshenko shell-type model of the local region is solved analytically for a set of smooth forces that will be applied to the full three-dimensional shell in place of the constraint. Third, the model of a full three-dimensional shell with end caps is analyzed in sonax, with the smooth forces applied on a mode-by-mode basis. The complete solution to the original problem is the superposition of the solutions of each of the subproblems described. Note that the high modal content of the original problem is contained in the high-resolution separate local problem (small domain), leaving only low modal content in the global (large domain) problem that is solved using sonax. This sample problem demonstrates the ability of ANM to enhance the capabilities of sonax and to improve the convergence rate (and overall computation time) for problems of this type. [Work sponsored by ONR.]

An illustration of analytical/numerical matching with finite-element analysis for structural vibration problems

Analytical/numerical matching (ANM) is an accurate and efficient method for solving many types of problems with discontinuities. The method separates local and global effects, and solves separate subproblems using high resolution around the discontinuity and low resolution away from the discontinuity. The work presented in this manuscript demonstrates a methodology for applying ANM to a dynamic structure using finite-element analysis (FEA) for the solution of the high-resolution (local) and the low-resolution (global) subproblems. The ANM method is illustrated on a thick, two-dimensional beam having several displacement constraints attached to its lower surface. Ordinarily (and here, for verification purposes) this problem would be solved using two-dimensional plane elements due to the local discontinuities around the constraints and the thickness of the beam. Using ANM, these discontinuities and through-thickness effects are modeled in the geometrically compact local problem using a high-resolution mesh of two-dimensional eight-node plane elements. The much larger global problem contains no discontinuities and is reduced to the solution of a low-resolution finite-element mesh of two-node Bernoulli–Euler beam elements. A third subproblem (matching) is solved analytically (no computational overhead). The agreement between the ANM solution and the purely FEA solution is excellent, and the computational savings are significant.

Structural-acoustic analysis of 3-D curved shell with a discontinuity using analytical/numerical matching (ANM)

Analytical/numerical matching (ANM) is used to determine the structural vibration of a curved shell driven at a single support. The ANM solution decomposes the problem into global, local, and matching subproblems. The global problem addresses large-scale effects with structural discontinuities replaced by smooth distributed forces. The local problem models the rapidly changing region around a structural discontinuity. These constituent problems are solved independently by the most efficient method available. Here, the local problem is solved numerically using 3-D solid finite element analysis (FEA) and the matching problem is solved analytically, using the Love–Timoshenko shell equations. The global problem is decomposed modally in azimuth, leaving a corresponding FEA problem in the axial direction for each mode. Due to the smoothness of the global problem, relatively few modes are required for the modal decomposition and the affiliated FEA problem can be solved using a low resolution. An advantage of the ANM method is its ability to capture the detailed response of structures with geometric complexities without having to incorporate the complexity (via higher-order elements or high mesh densities) in the modeling of the large global problem. [Work sponsored by ONR.]

Computationally efficient modeling of structural inhomogeneities by analytical/numerical matching

Solving structural acoustic problems in the mid-frequency range is a challenging task. Computation is expensive due to both the large well-coupled nature of the domain and resolution constraints such as small flexural wavelength. In addition, discrete structural elements such as ribs and stiffeners cause localized high-resolution content that requires much greater resolution than would be otherwise warranted. A method called analytical/numerical matching (ANM) alleviates this additional computational burden. The ANM method captures the particular influence of the inhomogeneity on the structure in an analytically expressed local solution. The governing equation is then used to show that the extraction of this local solution by superposition amounts to replacing the original discrete influence of the inhomogeneity by a smoothed forcing derived from the local solution. The result is a smoother problem that can be computed more efficiently and without the loss of any information. The method has been demonstrated for several configurations of increasing complexity in the process of its development as a useful tool for efficiently solving realistic problems of engineering interest. These problems have shown the usefulness, accuracy, and efficiency of the method. Examples will be presented, including recent work on implementation within a finite element analysis.

Scattering from fluid-loaded cylindrical shell with periodic circumferential constraints using analytical/numerical matching

Acoustic scattering from an infinitely long, thin elastic cylindrical shell with axially periodic circumferential constraints leads to the excitation of flexural, longitudinal, and shear waves on the shell. Of interest is the interaction of these structural waves with the discontinuous constraints, and the effect of this interaction on the resultant scattered sound field. This three-dimensional, fully coupled, structural acoustic problem is solved using a method called analytical/numerical matching (ANM). The method combines high-resolution local analytical solutions and low-resolution global numerical solutions to more efficiently model structural discontinuities. The ANM local solutions have been developed to efficiently capture the rapid variation in system response across the discontinuous constraint. The local solutions are combined with a smooth global solution, modeled by modal methods, to form an accurate, uniformly valid composite solution. The ANM composite solution is more accurate and converges much more rapidly than the traditional modal approach. An important observation is that this scattering problem exhibits considerable sensitivity to modeling accuracy and convergence of structural response in the region of the discontinuities. Errors in these regions have an overall effect on the structure and the associated scattered field.

Analytical/numerical matching applied to a finite beam with multiple nonperiodically spaced constraints

Analytical/numerical matching (ANM) is used to solve for structural vibrations of a finite beam constrained by multiple, nonperiodic supports. The ANM solution decomposes the stated problem into global, local, and matching problems. The global problem addresses large scale effects, with structural discontinuities replaced by smooth distributed forces. The local problem models the rapidly changing region around a structural discontinuity. These constituent problems are solved independently by the most efficient method available. Here, the local problem is solved numerically using finite element analysis (FEA) and the matching problem is solved analytically. The global problem is modeled using FEA with third-order Bernoulli–Euler beam elements. However, the solution procedure of the FEA model must be modified, in order to be compatible with ANM. Two-dimensional plane elements are required only in the local solution, to resolve the discontinuity. This finite-element order reduction provides significant savings in computation time. Comparison with traditional solution methods shows that the ANM approach accurately solves for local effects at each support, in addition to the overall response of the vibrating structure. The framework of applying ANM to more complex problems is also presented. [Work sponsored by ONR.]

Analytical/numerical matching for efficient calculation of scattering from cylindrical shells with lengthwise constraints

Structural discontinuities in highly coupled fluid–structure systems are modeled by an approach called analytical/numerical matching (ANM). This method separates the low-resolution global influence of a discontinuity from the relatively high-resolution local effects. A continuous, smoothed replacement for a fundamental structural discontinuity is constructed so that the system is identically unchanged beyond a small smoothing region. Simultaneously, the precise local effect of smoothing the discontinuity is retained in analytical form. The smoothed problem is solved by numerical techniques, with rapid convergence and reduced computational cost. The original discontinuous character is restored using the analytical expression for the local difference between the smoothed and the original problems. ANM has been successfully applied to two-dimensional cases of acoustic scattering from a thin, infinitely long cylindrical shell, with multiple structural discontinuities. Local solutions for longitudinal line discontinuities with radial, tangential, and rotational constraints have been formulated using ANM. Line constrained scattering problems, as well as line driven problems, are investigated. The ability of analytical/numerical matching to replace a discontinuous physical problem by a well behaved continuous one for numerical evaluation, while ultimately retaining the original geometry and physical behavior, is illustrated.

Novel Method for Calculating Two-Dimensional Blade Vortex Interaction

The methodology involves the application of a newly developed aeroacoustic boundary element method, which uses analytical/numerical matching (ANM), in conjunction with a novel vortex method, turbulent core model (TCM) vortex dynamics. ANM is a hybrid scheme combining a low-resolution global numerical solution with high-resolution local solutions to form a composite solution. ANM provides high-resolution calculations from low-resolution numerics with analytical corrections, while avoiding the subtlety involving singular integral equations and their numerical implementation.

The application of analytical/numerical matching to structural discontinuities in structural/acoustic problems

Analytical/numerical matching (ANM) is introduced to more efficiently model structural discontinuities in structural/acoustic problems. The general method is explained in detail for the simple example problem of normal incidence acoustic scattering from a periodically line-supported elastic membrane. A modal series solution exists for this problem. The most slowly convergent part of this solution, the dominant high-resolution (large wave number) content, is removed from the series representation and summed in closed form in what is called the ANM local solution. This local solution is represented by a piecewise polynomial that is zero outside of a small region. The remaining part of the solution, containing the overall low resolution content, is called the ANM global solution and converges very rapidly. In the present example, the series representing the global solution falls by three powers of the index faster than the classical modal solution. The result is an ANM composite (local plus global) solution that requires far less computational effort to resolve. In addition to providing a clear abstraction of the method, the example problem allows the equivalence of the ANM solution to be verified, and indicates the importance of resolving the fine scale effects of discontinuities in constrained scattering problems.

An acoustic boundary element method using analytical/numerical matching

Analytical/numerical matching (ANM) is a hybrid scheme combining a low-resolution global numerical solution with a high-resolution local solution to form a composite solution. ANM is applied to a harmonically oscillating body to calculate the radiated acoustic field and the associated fluid loading. The approach utilizes overlapping smoothed dipoles, and local corrections to calculate the dipole strength distribution along the surface of the body. A smoothing length scale is introduced that is larger than the smallest physical scale, and smaller than the largest physical scale. The global low-resolution solution is calculated numerically using smoothed dipole solutions to the wave equation, and converges quickly. Local corrections are done with high-resolution local analytical solutions. The global numerical solution is asymptotically matched to the local analytical solutions via a matching solution. The matching solution cancels the global solution in the near field, and cancels the local solution in the far field. The method is very robust, offering insensitivity to node location. ANM provides high-resolution calculations from low-resolution numerics with analytical corrections, while avoiding the usual subtleties involving singular integral equations, and their numerical implementation. The method is applied to calculate the radiated acoustic field and surface pressure of various flat plate configurations in two dimensions. An oscillating rigid flat plate, a forced elastic flat plate, plane-wave diffraction, and mechanical impedance calculations are addressed.

Novel approach to aerodynamic analysis using analytical/numerical matching

Novel approach to aerodynamic analysis using analytical/numerical matching