Direct Global Matrix Research Articles

Electromagnetic field produced by radiation source submerged in non-homogeneous seawater

Electromagnetic (EM) waves are one of the important carriers for information exchange in seawater. The EM parameters of seawater are important factors that affect the performance of EM wave propagation. In the real marine environment, non-constant EM parameters make it impossible to consider seawater as a homogeneous media in the model. However, there are few targeted analyses in the existing studies. In this paper, an N-layered media model was established to investigate the influence of non-constant EM parameters on EM wave propagation in seawater. A direct global matrix method is proposed to solve the EM fields, whose result accuracy does not decrease with the increasing number of layers. The necessity and correctness of the model and methods are verified through numerical simulations and sea experiments. Combined with the existing marine environment database, the propagation characteristics of EM waves in different EM parameter profiles of seawater were analyzed through numerical experiments. The results indicated that the non-homogeneous seawater has a great impact on the intensity, phase, change rate, and spatial distribution of EM waves. Furthermore, the influence is not only reflected in the underwater EM field but also in the air and increases with frequency.

Three-dimensional coupled mode analysis of internal-wave acoustic ducts.

A fully three-dimensional coupled mode approach is used in this paper to describe the physics of low frequency acoustic signals propagating through a train of internal waves at an arbitrary azimuth. A three layer model of the shallow water waveguide is employed for studying the properties of normal modes and their coupled interaction due to the presence of nonlinear internal waves. Using a robust wave number integration technique for Fourier transform computation and a direct global matrix approach, an accurate three-dimensional coupled mode full field solution is obtained for the tonal signal propagation through straight and parallel internal waves. This approach provides accurate results for arbitrary azimuth and includes the effects of backscattering. This enables one to provide an azimuthal analysis of acoustic propagation and separate the effects of mode coupled transparent resonance, horizontal reflection and refraction, the horizontal Lloyd's mirror, horizontal ducting and anti-ducting, and horizontal tunneling and secondary ducting.

Numerical Solution of Range-Dependent Acoustic Propagation

The direct global matrix approach can be applied to modeling of range-dependent sound propagation in order to achieve numerically stable and accurate solutions. By solving the global system directly, this method features high efficiency as well as accuracy by avoiding error accumulation. It is an important issue to solve linear systems numerically in the direct global matrix approach, especially for the large-scale problems. An efficient and memory-saving algorithm is developed for solving the global system, in which the global coefficient matrix is treated as a block pentadiagonal matrix. As a result, this numerical model has the ability to solve large-scale problems on regular computers. Numerical examples are also presented to demonstrate the accuracy and efficiency of this method.

A coupled-mode method for sound propagation in a range-dependent penetrable waveguide

The coupled-mode method based on the direct global matrix (DGMCM) approach for sound propagation in range-dependent waveguides [Luo et al., "A numerically stable coupled-mode formulation for acoustic propagation in range-dependent waveguides," Sci. China-Phys. Mech. Astron. 55, 572 (2012)] is further developed. The normal mode model KRAKEN is adopted to provide local modal solutions and their associated coupling matrices. As a result, the model DGMCM is capable of providing full two-way solutions for the two-dimensional realistic problems characterized by a penetrable bottom and a depth-varying sound speed profile. In addition, the closed-form expressions of coupling matrices for sound propagation in a range-dependent, two-layer waveguide are proposed. The numerical solutions of the coupling matrices by DGMCM agree well with the analytical solutions. Sound propagation in a penetrable wedge is solved by the updated DGMCM model. The numerical results indicate that the updated DGMCM model is numerically stable and accurate, and can provide benchmark solutions for realistic range-dependent problems.

A numerically stable coupled-mode formulation for acoustic propagation in range-dependent waveguides

In this paper, a stepwise coupled-mode model with the use of the direct global matrix approach is proposed. This method is capable of handling two-dimensional problems with either a point source in cylindrical geometry or a line source in plane geometry. With the use of the direct global matrix approach, this method is numerically stable. In addition, by introducing appropriately normalized range solutions, thismodel is free from the numerical overflow problem. Furthermore, we put forward source conditions appropriate for the line-source problem in plane geometry. As a result, this method is capable of addressing the scenario with a line source on top of a sloping bottom. Closed-form expressions for coupling matrices are derived and applied in this paper for handling problems with pressure-release boundaries and a homogeneous water column. The numerical simulations indicate that the proposed model is accurate, efficient, and numerically stable. Consequently, this model can serve as a benchmark model in range-dependent propagation modeling. Although this method is verified by an ideal wedge problem in this paper, the formulation applies to realistic problems as well.

Generalized Coupled-Mode Formulation for Sound Propagation in Range-Dependent Waveguides

An accurate and numerically stable method based on the coupled-mode theory is presented. By applying the direct global matrix approach to obtain the modal expansion coefficients, this method is numerically stable. In addition, appropriately normalized range solutions are introduced, which resolves the overflow problem entirely. Furthermore, we put forward source conditions appropriate for the line-source problem in plane geometry. As a result, this method is capable of dealing with the scenario where a line source is located inside the region of a deformation. Closed-form expressions for coupling matrices are provided for ideal waveguides. Numerical results indicate that the present method is accurate and numerically stable. Consequently, this model can serve as a benchmark in range-dependent propagation modeling.

Spherical wave effects in sediment reflection coefficients

Several research institutions are investigating models to properly describe high frequency, sandy-sediment, acoustic properties because of their importance in shallow water operations. In this talk we will focus on the error contribution of spherical wave effects on each of four common models: the visco-elastic model, the Effective Density Fluid Model (EDFM), Buckinghams micro-sliding model and the Biot/Stoll poro-elastic model. This will be accomplished by comparing the known models of reflection coefficients to laboratory measurements. Theoretical data will be calculated for each of the models using OASES, which utilizes the depth-dependent Green’s function and a direct global matrix approach to calculate the full field. Parameters for the visco-elastic model have been measured and verified at the test location. Parameters for the EDFM, Buckingham and Biot/Stoll models were determined from measured values and previous inversions of plane wave reflection data taken at the same location. Air/water interface data will be measured and computed for each model as a test case. Linearly Frequency Modulated (LFM) chirps over the range of 30–160 kHz will be used to establish ranges in frequency. [Work supported by ONR, Ocean Acoustics.]

Karhunen-Lo-ccedil;ve Decomposition of the Transient Dynamics of a Multibay Truss

We compute the transient dynamics of an impulsively loaded multibay truss. In computing the transient responses we avoid numerical instabilities due to exponential dichotomy of the bay transfer matrices by employing the direct global matrix approach. This method eliminates the need of transfer matrix multiplications that are prone to numerical instabilities and, instead, requires a numerically stable inversion of a global transfer matrix of large dimension. The transient responses at multiple points of the truss are used to perform system Identification of the dynamics using Karhunen-Loeve (K-L) decomposition. This analysis computes dominant coherent structures (proper orthogonal modes) of the truss under specific transient excitations, together with the energy of each identified coherent structure. The identified K-L modes can be used to develop accurate low-order models of the truss dynamics for control or substructure synthesis purposes.

Determination of elastic constants of composites by time-resolved acoustic microscopy

By using the angular spectrum acoustic microscopy model and the direct global matrix solution for a composite plate, the reflected signal from a time-resolved line focus transducer has been analyzed. The generalized ray solution has been used to decompose the complicated reflected signals into different wave modes. Strategies to determine the elastic constants using time-resolved acoustic microscopy have been discussed. Both the lateral wave and the bulk wave reflected from the bottom surface of a composite plate have been used in the elastic constant determination.

Karhunen-Loeve decomposition of the transient dynamics of a multibay truss

We compute the transient dynamics of an impulsively loaded multibay truss. In computing the transient responses we avoid numerical instabilities due to exponential dichotomy of the bay transfer matrices by employing the direct global matrix approach. This method eliminates the need of transfer matrix multiplications that are prone to numerical instabilities and, instead, requires a numerically stable inversion of a global transfer matrix of large dimension. The transient responses at multiple points of the truss are used to perform system Identification of the dynamics using Karhunen-Loeve (K-L) decomposition. This analysis computes dominant coherent structures (proper orthogonal modes) of the truss under specific transient excitations, together with the energy of each identified coherent structure. The identified K-L modes can be used to develop accurate low-order models of the truss dynamics for control or substructure synthesis purposes.

Vibration of truss structures

Recent trends in underwater vehicle design suggest the use of trusslike structures to support vibrating machinery. Experimental measurements are used to understand the dynamic behavior of a set of 1:15 model, three-dimensional truss structures over the full scale equivalent frequency range 10–1400 Hz. A cubic truss with all equal struts, a rectangular truss with two strut sizes, and a less practical truss composed of pyramid-shaped cells are considered. In conjunction with experiments, a direct global stiffness matrix numerical model is used to identify fundamental truss processes. The periodic nature of trusses causes them to act as mechanical comb filters in frequency with 3–5 dB per bay of attenuation at nonresonant frequencies. In trusses with long runs of axial struts, energy is carried down the truss axis primarily by compressional resonances. The spacing of the resonant peaks is controlled by strut length. A filtering strategy uses strut length to reduce the overall truss response via destructive interference of the frequency resonance structures of the individual struts. Multiple periodicities and impedance discontinuities at the truss joints cause the resonant peaks at low frequencies to widen. The consequence of this is that trusses with varied strut lengths tend to a low-pass frequency characteristic. This effect is compared to an equivalent applied damping and is shown to achieve up to 6 dB per bay of attenuation at the higher frequencies.

Acoustic radiation from a 3-D truss: Direct global stiffness matrix modeling results

This study was initiated to evaluate the normalized sound power level radiated by a driven truss. The coincidence frequency for the beam elements was sufficiently high that global truss modes are not a concern. Radiation from local beam modes dominate the field. Acoustic radiation was modeled using a combination of the direct global stiffness matrix (DGSM) method and models based on radiation efficiency of cylindrical beams. Use of a nominal structural loss factor of 10−4 for the beam material (aluminum) overestimates the measured field by an order of magnitude. Earlier studies suggest that a combination of structural loss in the joints and multipath wave-type conversion in the truss leads to a loss factor of order 10−3. For this study experimental data were inverted to estimate a frequency-dependent loss factor, which confirms the prior estimate of order 10−3. When the DGSM based model was run with the estimated frequency-dependent loss factor, the model results match the measured data closely. A power balance shows that the structural loss factor dominates the total power dissipated in the system, as one might expect for a lightly fluid-loaded structure. [Research sponsored by ARPA/ONR.] a)E-mail: fricke@mit.edu

Direct global stiffness matrix method for thin cylindrical shell dynamics

This paper introduces the direct global stiffness matrix (DGSM) method for analyzing the dynamics of cylindrical shells with axial symmetry. The dynamic stiffness matrix of a straight, circular cross-section cylindrical element is formulated in terms of ring forces and displacements at each end on a per circumferential mode basis. The stiffness matrix is based on an exact wave-type solution of the thin shell Donnell equations. More complicated shell systems are analyzed by assembling any number of cylindrical elements. For example, a cylinder with a gradual variation in radius may be modeled by a series of shell elements welded together at their ends. Total system dynamics are found at any given frequency by inverting the global system matrix formed from the individual stiffness element matrices and summing over circumferential mode number. Results for a test case are presented to demonstrate and verify the implementation. [Research sponsored by ONR.]

Spectral super-element approach to range-dependent ocean acoustic modeling

The spectral super-element approach uses a hybridization of finite elements, boundary integrals, and wave-number integration to solve the Helmholtz equation in a range-dependent ocean environment. The range-dependent ocean is divided into range-independent sectors or super-elements. Wave-number integral representations can be derived for an influence matrix representing the relation between displacement and pressure expansions on the vertical boundaries. The integration kernels are determined very efficiently by the direct global matrix method in combination with a numerical quadrature scheme. The unknown expansion coefficients are then found by matching the boundary conditions of continuous displacement and stress between the sectors, with the acoustic field following by evaluating the wave-number integrals within each sector, e.g. using the FFP technique. Two different boundary matching schemes are applied. One solves the global coupling problem incorporating all multiple scattering. A second, more efficient approach applies a single-scatter approximation at each vertical sector boundary, allowing for a marching solution. Numerical solutions to a series of canonical benchmark problems are given and compared to solutions obtained by other numerical approaches such as parabolic equations and coupled modes.

Low‐frequency underwater transducer modeling using the direct global stiffness method

The direct global stiffness matrix method (DGSM), described by Fricke and Hayner [direct global stiffness matrix method for 3‐D Truss Dynamics, submitted to the ASME 15th Biennial Conference on Mechanical Vibration and Noise, 17–21 September 1995], provides an efficient method for analyzing two‐dimensional and axisymmetric, low‐frequency, underwater, transducer geometries. The DGSM method is a wave‐type solution based on the Euler–Bernoulli beam element. The mechanical motion of the transducer element is modeled using a collection of beam elements welded together at joints. By imposing dynamic equilibrium conditions at these joints, a banded symmetric stiffness matrix is formed. The matrix is then solved using Gaussian elimination to find joint displacements and rotations with beam energies calculated on a post‐processing basis. The radiation load due to the surrounding water is included using a compact source assumption. Thus fluid loading interaction effects, important for low quality factor transducers, are properly addressed. Analysis of a class IV flextensional transducer is offered as an example of the usefulness of this approach.

Scattering from a multi-layered cylindrical shell containing a bulkhead

A multi-layered cylindrical shell is modeled with the direct global matrix approach (DGM). The DGM approach allows the shell to have multiple viscoelastic layers, which are modeled with the full 3-D equations of elastodynamics. The DGM formulation is numerically stable for the wave types of interest (membrane compression, membrane shear, and flexural waves). The shell also contains a thin elastic bulkhead, which is modeled with the Mindlin thin plate theory. This coupled system is excited by obliquely incident plane waves in a fluid medium that surrounds the shell. The scattering caused by the bulkhead is computed. The effects of viscoelastic layers are investigated. [Work supported by ONR.]

Structural acoustics of cylindrical shells having multiple layers using the direct global matrix

A method has been developed to compute the response of cylindrically layered viscoelastic systems using the direct global matrix approach (DGM). The numerical stability of the method allows us to model multilayered shells that have viscoelastic layers for damping and compliant coatings for decoupling. In this presentation, the method is applied to a cylindrical shell having external fluid loading and a viscoelastic layer of damping material sandwiched between two solid layers. The model can be excited by ring forces, point forces, and a point source in the fluid. Parameter studies are presented to investigate the effect of the viscoelastic damping layer on the various wave types (flexural, membrane compression, and membrane shear). [Work supported by ONR.]

A Spectral Element Approach to Wave Motion in Layered Solids

A matrix methodology similar to that of the finite element method is developed for the analysis of stress waves in layered solids. Because the mass distribution is modeled exactly, the approach gives the exact frequency response of each layer. The fast Fourier transform and Fourier series are used for inversion to the time/space domain. The impact of a structured medium with multiple layers is used to demonstrate the method. Comparison with existing propagator and direct global matrix methods show the present approach to be computationally more efficient.

Computational aspects of synthetic seismograms for layered media

A modified direct global matrix method for computation of synthetic seismograms in a layered viscoelastic media has been developed by combining information obtained from continuous and discrete formulations. The resulting code is stable for vertical seismic profiling as well as normal horizontal profiling and has proven efficient in the generation of full-scale seismograms. The wave field is first computed in the wave slowness domain. A proper integration back to the frequency domain is a critical part of a forward modeling code. By considering the discrete wavenumber formulation it is shown that numerical integration with equally spaced integration points corresponds to a fictitious cylindrical reflective surface with diameter inversely proportional to the sampling interval. It is demonstrated that for large ranges Filon integration is a natural extension. It is demonstrated that a trapezoidal Filon integration is more accurate than higher-order Filon schemes, which generate earlier spurious arrivals. In order to yield seismograms without spurious arrivals, the wavenumber or slowness sampling must be sufficiently dense. However, in some regions, the reflectivity function is quite regular. In these regions, linear interpolation is adequate and the number of sampling points is reduced by an adaptive scheme. Finally, in the slowness domain as opposed to the wave-number domain, the position of the peaks in the reflectivity function is nearly independent of frequency. This enables an efficient vectorization over frequency.

Global matrix formulation of wave phenomena in plane layered media

A direct global matrix formulation for computing the multilayer Green's impulse response function in media with arbitrary depth-dependent elastic or fluid properties is reniewed, after recent work by Schmidt. The global Green's function matrix for solving the resulting linear system of elastodynamic equations is formally analogous to the global stiffness matrix in the direct Finite Element Method and proves both computationally efficient and numerically stable. This analogy and the matrix solution are discussed, as embodied in the general applications code, “SAFARI” ( ∗ SA clant FA st Field Program for R ange- I ndependent Environments ∗ ), developed by one of us (H.Schmidt). Cros s-disciplinary applications of this direct global matrix algorithm are considered for three pertinent areas of current research; namely underwater acoustic propagation, ultrasonic scattering, and crustal seismic modeling. Results obtained show that the SAFARI model correctly reproduces the complete wavefield response in plane-layered, welded, homogeneous media. Arbitrary source/receiver numbers, frequencies, locations, and geometries can also be considered over much larger-scale multilayered media models than previously feasible with extant propagator matrix-based codes. The physical simplicity, CPU savings, and flexible performance of the direct global matrix algorithm suggests that many more full wavefield phenomena can now be simulated without excessively complicated, specialized, or expensive modeling schemes.