Energy Finite Element Analysis Research Articles

Energy Finite Element Analysis for Structural-Acoustic Design of Naval Vehicles

The pursuit of innovative ship designs, the use of new materials, and the increasingly demanding performance requirements dictate a broadly focused thrust for addressing noise and vibration issues during the design process. This expanded technical view also requires that the types of tools used for these assessments can provide results within a short enough time to affect the design cycle. Recently, an Energy Finite Element Analysis (EFEA) formulation has been developed for computing the vibration and the associated radiated noise of complex vehicles at frequencies where conventional finite element methods are no longer efficient. In this paper the main theoretical aspects of the EFEA are presented first. Next, a substructuring capability is described where large computational models, typical of naval applications, can be subdivided into separate domains for pursuing a compartmentalized solution and achieving further improvements in computational efficiency. Finally, validation case studies are presented with correlation between computations and scaled model testing.

Energy Finite Element Analysis for Structural-Acoustic Design of Naval Vehicles

The pursuit of innovative ship designs, the use of new materials, and the increasingly demanding performance requirements dictate a broadly focused thrust for addressing noise and vibration issues during the design process. This expanded technical view also requires that the types of tools used for these assessments can provide results within a short enough time to affect the design cycle. Recently, an Energy Finite Element Analysis (EFEA) formulation has been developed for computing the vibration and the associated radiated noise of complex vehicles at frequencies where conventional finite element methods are no longer efficient. In this paper the main theoretical aspects of the EFEA are presented first. Next, a substructuring capability is described where large computational models, typical of naval applications, can be subdivided into separate domains for pursuing a compartmentalized solution and achieving further improvements in computational efficiency. Finally, validation case studies are presented with correlation between computations and scaled model testing.

Energy Finite Element Analysis Developments for Vibration Analysis of Composite Aircraft Structures

The Energy Finite Element Analysis (EFEA) has been utilized successfully for modeling complex structural-acoustic systems with isotropic structural material properties. In this paper, a formulation for modeling structures made out of composite materials is presented. An approach based on spectral finite element analysis is utilized first for developing the equivalent material properties for the composite material. These equivalent properties are employed in the EFEA governing differential equations for representing the composite materials and deriving the element level matrices. The power transmission characteristics at connections between members made out of non-isotropic composite material are considered for deriving suitable power transmission coefficients at junctions of interconnected members. These coefficients are utilized for computing the joint matrix that is needed to assemble the global system of EFEA equations. The global system of EFEA equations is solved numerically and the vibration levels within the entire system can be computed. The new EFEA formulation for modeling composite laminate structures is validated through comparison to test data collected from a representative composite aircraft fuselage that is made out of a composite outer shell and composite frames and stiffeners. NASA Langley constructed the composite cylinder and conducted the test measurements utilized in this work.

Effects of imperfect adhesion on thermal micro-residual stresses in polymer matrix composites

This paper investigates the influences of imperfect bonding between the fiber and matrix on thermal micro-residual stress fields in polymer matrix composites. For this purpose, a representative volume element consisting of a three-phase composite material subjected to a uniform temperature change is considered. Based on the energy method, a three-dimensional closed-form solution for micro-residual stresses is obtained. Besides, a finite element model is developed and the results are compared with the analytical solution. Both the energy method and finite element analysis show similar trend for thermal stress distribution along the fiber length, while due to the stress singularity, the interfacial shear stress from the finite element solution cannot satisfy the stress-free condition at the fiber end. The analysis shows that the magnitude of thermal stresses and their distribution mainly depend on the bonding efficiency parameter. An increase in thermal and elastic properties bonding efficiencies leads to a considerable decrease in composite axial and shear residual stresses, while the Poisson's ratio bonding efficiency does not affect the thermal stress fields. The interfacial radial residual stress distribution is approximately independent of the bonding conditions. Inefficient bonding may result in higher residual stresses in comparison with the perfect bonding condition. It means that in cases of low bonding efficiency conditions, the ability of composites to sustain and transmit load decreases drastically. Thermal stress concentration occurs at the vicinity of the fiber ends, although peak values depend on the bonding efficiency value.

Transient energy density distribution of a rod under high-frequency excitation

Energy finite element analysis (EFEA) is an efficient way to solve high-frequency structural dynamics response problems. Up to now, EFEA has been used to deal with time-independent vibration problems. However, it is still necessary to understand the time dependent details of energy density distribution of a structure. To study the transient response of a rod under high-frequency sinusoidal excitation, the transient energy density governing equation for a rod is presented. The governing equation is solved, and the solution is verified using an analytical method. Example application to a rod is presented to illustrate the feasibility.

An Investigation of Reverberation Chamber High-Frequency Vibration-Acoustic Problem for an Aircraft Cabin Title

In this paper, reverberation chamber high-frequency vibration-acoustic problem for an aircraft cabin is investigated with two methods. One is combining energy finite element analysis (EFEA) and energy boundary element analysis (EBEA); another is boundary element method by SYSNOISE. Comparison is performed between these two methods.

Energy finite element analysis of naval vehicles.

Simulation methods can be used for meeting both the new acoustics and vibration initiatives for surface ship silencing. An energy finite element analysis (EFEA) formulation has been developed for computing efficiently the vibration of complex structures and the associated radiated noise at frequencies beyond the limits of conventional finite element methods. The EFEA provides a very attractive alternative to the statistical energy analysis (SEA) because it uses the same finite element model which is utilized for other simulations (i.e., structural analysis), while the SEA requires the development of a completely new model based on a lumped parameter approach. Further, the SEA requires estimates of modal densities for the various subsystems which introduce difficulties and approximations in the solution. Thus, the EFEA streamlines the structural-acoustic analysis within the ship design process due to modeling commonality with other disciplines. In this presentation, the main theoretical aspects of the EFEA will be presented first. Results presented for two 1/8th scale structures representing an advanced double hull design and a conventional hull design of a surface ship are analyzed. Results for the structural vibration induced on the outer bottom part of the structures are compared to available test data.

A Three-dimensional Finite Element Method for Surface Effects

Due to the relatively high surface-to-volume ratio, the surface effect can be significant for nanoscale materials. A numerical method, which combines surface energy and three-dimensional (3-D) finite element analysis, is proposed to simulate the elastic and plastic deformation of materials and structures at nanoscale. To demonstrate the method is valid and efficient, the free relaxation of single crystalline Cu nanowires is investigated and the numerical results are compared to the atomic simulation results.

Energy finite energy analysis for shipboard noise

Machinery-induced habitability noise is difficult to model efficiently and accurately. The potential of energy finite-element analysis (EFEA) is compared to other prediction tools such as statistical energy analysis (SEA). This paper will explore the benefits and costs of EFEA with respect to SEA for acoustic modeling. The focus will be on issues relating to structural modeling for EFEA purposes. EFEA techniques will be evaluated to see if they possess the capabilities of verified SEA approaches for predicting habitability and radiated noise, where it is necessary to account for the impact of diverse marine constructions and sources such as the lack of machinery source information with respect to force or moment inputs or the finite impedance of machinery foundations. The effort proposed herein will provide the necessary engineering to research and identify salient features of EFEA that are potentially applicable for the detailed analysis of the acoustic environment and response of surface ships to various excitation sources. The paper will also address the pros and cons of SEA versus energy-finite element analysis (EFEA) methods used to predict the habitability noise of surface ship platforms. [This work is supported by an Office of Navy Research contract.]

Recent developments and applications of energy finite- element analysis

The energy finite-element analysis (EFEA) comprises a finite-element-based solution for vibration and acoustic modeling that uses energy-based variables for formulating the governing differential equations. This presentation covers the main points in the derivation of energy finite-element analysis (EFEA), validation through comparison with test data, and case studies that demonstrate the computational capabilities of the EFEA. The EFEA can be applied in various engineering disciplines as it will be demonstrated by the case studies. Thus, it comprises a general purpose simulation method for vibration and acoustic analysis of complex systems. [Work support for this research is provided by ONR code 331.]

Energy finite-element analysis of the NASA aluminum test-bed cylinder

A formulation was developed in the energy finite-element analysis (EFEA) for modeling the vibration of cylindrical structures with periodic axial and circumferential stiffeners. Appropriate power transfer coefficients are computed from the values of the propagation constants. The joint matrices of the EFEA formulation are computed based on the power transfer coefficients derived from periodic structure theory. EFEA analyses are performed for the NASA aluminum test-bed cylinder and simulations are compared to experiments. Excitation is applied on the test-bed cylinder by four shakers and the vibration is measured at 40 bays formed by axial and circumferential stiffeners. The EFEA results are compared successfully to test data between the 1/3 octave bands of 315 and 6300 Hz. [Work sponsored by NASA Langley Structural Acoustics Branch.]

Numerical investigation of small-scale surface ships by energy finite-element analysis

Experimental investigations have been performed to understand the structural acoustic characteristics of two small-scale surface ship models representing sections of conventional and advanced double-hull ship designs [Craun et al., ‘‘Experimental and analytical investigation of surface shiphull structure acoustics using physical models,’’ J. Acoust. Soc. Am. 116, 2569 (2004)]. Recently, new developments have been added to the energy finite-element analysis (EFEA) for surface ships [Wu and Vlahopoulos, ‘‘Vibratory response of surface ships predicted by energy finite element approach,’’ J. Acoust. Soc. Am. 116, 2569 (2004)]. The presentation will compare hull responses from EFEA predictions to test data for the two small-scale models to validate the new development in EFEA. Furthermore, the impact of the hull responses due to the different model designs will be discussed. [Support for this research is provided by ONR code 334.]

Combining an energy boundary element analysis with an energy finite element analysis for computing airborne noise inside a flexible structure due to an exterior acoustic source

The energy boundary element analysis (EBEA) has been utilized in the past for computing the exterior acoustic field at high frequencies (above ∼400 Hz) around vehicle structures and numerical results have been compared successfully to test data [Wang et. al., J. Sound Vib. 126, 278–283 (2004); Internoise (2002)]. The EFEA has been developed for computing the structural vibration of complex structures at high frequencies and validation has been presented [Zhang et. al., J. Sound Vib. 282, 679–700 (2005); 41, 11–12 (2005); Finite Elements Anal. Des. 41(11-12) (2005); Wang et. al., SAE Paper No. 2005-01-2372]. In this presentation the EBEA is utilized for computing the acoustic field around a cylindrical structure (similar to the NASA aluminum test-bed cylinder) from an external acoustic noise source. The computed exterior acoustic field comprises the excitation for the EFEA analysis. The structural vibration and the interior noise levels are computed, demonstrating how the EBEA and the EFEA can be combined in airborne noise applications for performing a complete vibro-acoustic analysis. [Work supported by ONR Code 334, SBIR program.]

A vibration analysis of stiffened plates under heavy fluid loading by an energy finite element analysis formulation

An energy finite element analysis (EFEA) formulation for high frequency vibration analysis of stiffened plates under heavy fluid loading is presented. The added mass effect and the radiation dampin...

A vibration analysis of stiffened plates under heavy fluid loading by an energy finite element analysis formulation

An energy finite element analysis (EFEA) formulation for high frequency vibration analysis of stiffened plates under heavy fluid loading is presented. The added mass effect and the radiation damping effect due to the fluid loading are incorporated in the derivation of the EFEA governing differential equations. In addition, the fluid loading effect and the presence of a stiffener are accounted when computing the power transfer coefficients through plate/stiffener junctions. The effect of the fluid on power transfer coefficients of a plate/stiffener junction is analyzed and discussed. The new formulation is validated by comparing EFEA results to solutions by very dense finite element models for several plate/stiffener assemblies, which include two coplanar plates connected by a stiffener, two plates connected at a 45° angle by a stiffener, and a section of a typical naval structure. All plates are considered in contact with dense fluid from one side. The energy ratios between the receiving sections of the plates and the excited sections of the plates are computed and good correlations are observed between the EFEA and the very dense FEA results.

An energy finite element formulation for high-frequency vibration analysis of externally fluid-loaded cylindrical shells with periodic circumferential stiffeners subjected to axi-symmetric excitation

A hybrid method is developed for predicting the high-frequency vibration response of fluid-loaded cylindrical shells with periodic circumferential stiffeners. In this method, the cylindrical shell is modeled using the Energy Finite Element Analysis (EFEA) method which includes added mass and radiation effects due to the surrounding exterior fluid medium. The joint matrices of the EFEA formulation at the location of the periodic stiffeners are computed based on Periodic Structure (PS) theory. Thus, the periodicity effects such as pass/stop characteristics are captured in the EFEA solution. The hybrid EFEA-PS method is used to analyze the vibration of a fluid-loaded axisymmetric cylindrical shell with periodic circumferential stiffeners. The flexural energy stored in each periodic section is computed. The results are compared with the solution produced by a very dense axisymmetric Finite Element (FE) model with infinite finite elements for the fluid domain. The good correlation indicates that the new hybrid method captures properly both the heavy fluid effects of the exterior fluid medium and the periodicity characteristics due to periodic stiffeners.

Vibratory response of surface ships predicted by energy finite element approach

Energy finite element analysis (EFEA) and energy boundary element analysis (EBEA) have been successfully developed and validated for a submerged plate and a cylindrical shell (Zhang et al., J. Sound Vib. 263, 21–46) and have been applied to calculate flow noise of a submerged cylindrical shell under turbulent boundary layer excitation (K. Wu and N. Vlahopoulos, ASA 143rd Meeting). Recently, new capabilities are under development to extend EFEA/EBEA formulations for surface ship applications. The presentation will discuss the developmental work and present some numerical examples in validating the development. [Work Support for this research is provided by ONR Code 334.]

Development of an energy boundary element formulation for computing high-frequency sound radiation from incoherent intensity boundary conditions

An energy boundary element analysis (EBEA) formulation is presented for calculating sound radiation at high frequency from a radiator with arbitrary shape. The EBEA development presented in this paper compliments previous energy finite-element analysis (EFEA) developments for computing high-frequency vibration of structures immersed in an acoustic medium. From the EFEA solution the structural vibrational energy throughout the structure and the acoustic power radiated from each surface element in contact with the surrounding fluid are computed. The EBEA utilizes the information for the acoustic power radiated from each element of the outer surface of the structure as a boundary condition in order to compute the acoustic field in an unbound medium with no damping that surrounds the radiating structure. Since the structural vibration between different elements is considered as incoherent in the EFEA, the acoustic intensity associated with each surface element that radiates in the surrounding medium is also considered as incoherent between different elements in the EBEA formulation. The governing integral equation of EBEA is established, and a numerical solution is presented. The present method is validated by comparing EBEA results to analytical solutions and to numerical results from an axi-symmetric hp finite-element code with infinite acoustic elements suitable for structural acoustic computations. Good correlation is demonstrated for all the analyses.

A basic hybrid finite element formulation for mid-frequency analysis of beams connected at an arbitrary angle

When beams are connected at an arbitrary angle and subjected to an external excitation, both longitudinal and bending waves are generated in the system. Since longitudinal wavelengths are considerably longer than bending wavelengths in the mid-frequency region, the number of bending wavelengths in the beams is considerably larger than the number of longitudinal wavelengths. In this paper, plannar beams connected at arbitrary angles are considered. The energy finite element analysis (EFEA) is employed for modelling the bending behavior of the beams and the conventional finite element analysis (FEA) is utilized for modelling the longitudinal vibration in the beams. Thus, a basic hybrid FEA formulation is presented for mid-frequency analysis of systems that contain two types of energy. The bending vibration is associated with the long members in the system and the longitudinal vibration is associated with the short members. The long members are considered to have high modal overlap and to contain several wavelengths within their dimension, and uncertainty effects are present. The short members contain a small number of wavelengths, and exhibit a low modal overlap. Due to the low modal overlap the resonant frequencies are spaced far apart in the frequency domain, therefore the short members exhibit resonant or non-resonant behavior depending on the frequency of the excitation. In this work, the bending and the longitudinal vibration within the same beam member are treated as a long and as a short member, respectively. A hybrid joint formulation is developed between long and short members. Power reflection and transmission coefficients are derived for each joint. The distribution of the energy throughout the system demonstrates a strong dependency on the power transfer coefficients. Several systems are analyzed by the hybrid FEA and by analytical solutions, and good correlation between them is observed.

High-frequency vibration analysis of thin elastic plates under heavy fluid loading by an energy finite element formulation

An energy finite element analysis (EFEA) formulation for computing the high frequency behavior of plate structures in contact with a dense fluid is presented. The heavy fluid loading effect is incorporated in the derivation of the EFEA governing differential equations and in the computation of the power transfer coefficients between plate members. The new formulation is validated through comparison of EFEA results to classical techniques such as statistical energy analysis (SEA) method and the modal decomposition method for bodies of revolution. Good correlations are observed and the advantages of the EFEA formulation are identified.