Finite Element-transfer Matrix Research Articles

Time-harmonic analysis of acoustic pulsation in gas pipeline systems using the Finite Element Transfer Matrix Method: Theoretical aspects

Structural vibration of gas pipeline systems can be strongly affected by the response of the acoustic domain represented by the gas being transported through pressurized pipes, cylinders and other related components. The resultant Acoustically Induced Vibration (AIV) can be a cause of failure, for example, in reciprocating compressor systems, where the discharge and suction pulsations are the main source of excitation. The objective of this work is to explore and enhance the use of the Finite Element Transfer Matrix Method (FETM) for solving time-harmonic acoustic problems related to AIV, whose results can be used to analyze pulsation in piping systems. Four different damping models are explored and implemented (apart from the classical undamped model) resulting in five different acoustic elements for the quiescent fluid case. In addition, two different damping models are analyzed for the case of a fluid with mean flow, and implemented through two flow-acoustic elements. The physical behavior of certain typical junctions is also investigated, represented by acoustic and flow-acoustic length corrections, and an FETM formulation is proposed to account for local non-linear effects in transfer impedance elements. Also, a discussion related to the minimum and maximum FETM element sizes is presented. A critical analysis of the effectiveness of the whole FETM strategy is performed using a variety of representative gas pipeline geometries as application examples with a particular set of boundary conditions for quiescent and moving fluid medium. Nodal pressure frequency plots of cases considering all the presented acoustic and flow-acoustic elements, and related damping models, are presented. Coherent results were achieved using the procedure when frequency plots are compared with experimental data and 3D FEM equivalent models.

Obtaining the Transfer Function in One Dimensional Soil Amplification Analysis with Modified Finite Element Transfer Matrix Method

Bu çalışmada tek boyutlu zemin büyütme analizlerinde kullanılan Transfer fonksiyonlarının elde edilmesi için bir yöntem önerilmiştir. Çalışma kapsamında daha önce literatürde zemin hâkim periyodunun bulunması için kullanılan Değiştirilmiş Sonlu Elemanlar Taşıma Matrisi (DSETM) yöntemi zemin büyütme analizlerinde kullanılan transfer fonksiyonlarının bulunması problemine uyarlanmıştır. Klasik sonlu elamanlar yönteminde eleman sayısına bağlı olarak artan sistem matrisi DSETM yönteminde eleman sayısından bağımsızdır. Çalışmanın sonunda önerilen yöntemin uygunluğu iki örnek üzerinde gösterilmiştir. Ele alınan örnekler hem bu çalışmada sunulan DSETM yöntemiyle hem de literatürde yer alan yöntemle çözülerek sonuçlar karşılaştırılmıştır. Sunulan yöntemin uygulanması için Scilab ortamında bir program yazılmıştır.

Hybrid finite element transfer matrix method and its parallel solution for fast calculation of large-scale structural eigenproblem

Vibration characteristics play a fundamental role in structural analysis and dynamics design. The determination of vibration characteristics requires the solution of an eigenproblem. In this work, a novel parallel hybrid finite element (FE) transfer matrix method (TMM) is studied for a large-scale structural eigenproblem. The full structure is divided into some substructures, and the vibration characteristics of each substructure are calculated with their proper solution method (FE or TMM). Then by using the order-reduction transformation of the full structure based on an improved fixed interface modal synthesis, the reduced global equation of the full structure is obtained easily, and it can be used to solve the vibration characteristics of the full structure efficiently. To overcome the numerical instability of the ordinary transfer matrix technology in a large-scale structural analysis, the Riccati transform is also adopted. This hybrid method has good parallelism, and the corresponding parallel algorithms are presented. Finally, this method is applied to a vacuum vessel structure of the China fusion engineering test reactor, which has cyclic symmetry conditions. The simulation results demonstrate that this method has low computational cost and satisfactory accuracy.

AN APPLICATION OF THE MODIFIED FINITE ELEMENT TRANSFER MATRIX METHOD FOR A HEAT TRANSFER PROBLEM

The Transfer Matrix Method is an effective method which is used in mechanics and provides a great advantage especially in solving some mechanical problems. The most important advantage of the Transfer Matrix Method is reducing the dimensions of the matrix used in the analysis. The most important consequence of this is the saving the time. In this study, firstly studies in recent years in relation with the Transfer Matrix Method have been briefly summarized. Then the Modified Finite Element Transfer Matrix Method is explained briefly. At the end of the study, the application of the Modified Finite Element Transfer Matrix Method to the heat transfer problem has been demonstrated.

Efficient modeling of flat and homogeneous acoustic treatments for vibroacoustic finite element analysis. Finite size correction by image sources

This work is concerned with the hybrid finite element-transfer matrix methodology recently proposed by the authors. The main assumption behind this hybrid method consists in neglecting the actual finite lateral extent of the acoustic treatment. Although a substantial increase of the computational efficiency can be achieved, the effect of the reflected field (i.e. finite size effects) may be sometimes important, preventing the hybrid model from giving quantitative meaningful results. For this reason, a correction to account for wave reflections at the lateral boundaries of the acoustic treatment is sought. It is shown in the present paper that the image source method can be successfully employed to retrieve such finite size effects. Indeed, such methodology is known to be effective when the response of the system is a smooth function of the frequency, like in the case of highly dissipative acoustic treatments. The main concern of this paper is to assess accuracy and feasibility of the image source method in the context of acoustic treatments modeling. Numerical examples show that the performance of the standard hybrid model can be substantially improved by the proposed correction without deteriorating excessively the computational efficiency.

Efficient modeling of flat and homogeneous acoustic treatments for vibroacoustic finite element analysis. Direct field formulations

This paper is concerned with the development of a simplified model for noise control treatments to speed up finite element analysis in vibroacoustic applications. The methodology relies on the assumption that the acoustic treatment is flat and homogeneous. Moreover, its finite lateral extent is neglected. This hypothesis is justified by short wavelength and large dissipation, which suggest that the reflected field emanating from the acoustic treatment lateral boundaries does not substantially affect its dynamic response. Under these circumstances, the response of the noise control treatment can be formally obtained by means of convolution integrals involving simple analytical kernels (i.e. Green functions). Such fundamental solutions can be computed efficiently by the transfer matrix method. However, some arbitrariness arises in the formulation of the mathematical model, resulting in different baffling conditions at the two ends of the treatment to be considered. Thus, the paper investigates the possibility of different formulations (i.e. baffling conditions) within the same hybrid finite element-transfer matrix framework, seeking for the best strategy in terms of tradeoff between efficiency and accuracy. Numerical examples are provided to show strengths and limitations of the proposed methodology.

Free vibration analysis of axisymmetric shells with various shapes using Sylvester-transfer stiffness coefficient method

We developed a computational method for effectively conducting the free vibration analysis of axisymmetric shells with various shapes. This paper describes a computational algorithm for the free vibration analysis of axisymmetric shells using the Sylvester-transfer stiffness coefficient method (S-TSCM). From the free vibration analyses of three axisymmetric shells (joined conical-cylindrical shell, hermetic capsule and built-up shell), we verified the applicability of the S-TSCM. We then confirmed that the computational power of the S-TSCM is much better than both the finite element-transfer matrix method and finite element-transfer stiffness coefficient method, in terms of computational accuracy and time. In particular, when axisymmetric shells are modeled into a large number of conical shell elements, the S-TSCM is superior to both the bisection method using Sturm sequence property and Jacobi method, in terms of computational time and storage.

A hybrid finite element-transfer matrix model for vibroacoustic systems with flat and homogeneous acoustic treatments.

Practical vibroacoustic systems involve passive acoustic treatments consisting of highly dissipative media such as poroelastic materials. The numerical modeling of such systems at low to mid frequencies typically relies on substructuring methodologies based on finite element models. Namely, the master subsystems (i.e., structural and acoustic domains) are described by a finite set of uncoupled modes, whereas condensation procedures are typically preferred for the acoustic treatments. However, although accurate, such methodology is computationally expensive when real life applications are considered. A potential reduction of the computational burden could be obtained by approximating the effect of the acoustic treatment on the master subsystems without introducing physical degrees of freedom. To do that, the treatment has to be assumed homogeneous, flat, and of infinite lateral extent. Under these hypotheses, simple analytical tools like the transfer matrix method can be employed. In this paper, a hybrid finite element-transfer matrix methodology is proposed. The impact of the limiting assumptions inherent within the analytical framework are assessed for the case of plate-cavity systems involving flat and homogeneous acoustic treatments. The results prove that the hybrid model can capture the qualitative behavior of the vibroacoustic system while reducing the computational effort.

Erratum: Assessment of a hybrid finite element-transfer matrix model for flat structures with homogeneous acoustic treatments [J. Acoust. Soc. Am. 135, 2694–2705 (2014)

First Page

Assessment of a hybrid finite element-transfer matrix model for flat structures with homogeneous acoustic treatments.

Modeling complex vibroacoustic systems including poroelastic materials using finite element based methods can be unfeasible for practical applications. For this reason, analytical approaches such as the transfer matrix method are often preferred to obtain a quick estimation of the vibroacoustic parameters. However, the strong assumptions inherent within the transfer matrix method lead to a lack of accuracy in the description of the geometry of the system. As a result, the transfer matrix method is inherently limited to the high frequency range. Nowadays, hybrid substructuring procedures have become quite popular. Indeed, different modeling techniques are typically sought to describe complex vibroacoustic systems over the widest possible frequency range. As a result, the flexibility and accuracy of the finite element method and the efficiency of the transfer matrix method could be coupled in a hybrid technique to obtain a reduction of the computational burden. In this work, a hybrid methodology is proposed. The performances of the method in predicting the vibroacoutic indicators of flat structures with attached homogeneous acoustic treatments are assessed. The results prove that, under certain conditions, the hybrid model allows for a reduction of the computational effort while preserving enough accuracy with respect to the full finite element solution.

Free vibration analysis of asymmetric shear wall-frame buildings using modified finite element-transfer matrix method

In this study, the modified finite element- transfer matrix methods are proposed for free vibration analysis of asymmetric structures, the bearing system of which consists of shear wall-frames. In the study, a multi-storey structure is divided into as many elements as the number of storeys and storey masses are influenced as separated at alignments of storeys. The shear walls and frames are assumed to be flexural and shear cantilever beam structures. The storey stiffness matrix is obtained by formulating the governing equation at the center of mass for the shear walls and the frames in the i.th floor. The system transfer matrix is constructed in the dimension of 6x6 by transforming the obtained stiffness matrix. Thus, the dimension, which is 12nx12n in classical finite elements, is reduced to the dimension of 6x6. To study the suitability of the method, the results are assessed by solving two examples taken from the literature.

Modified finite element-transfer matrix method for the static analysis of structures

In this paper the Modified Finite Element-Transfer Matrix Method, which is the combination of Transfer Matrix Method and Finite Element Method, is applied to the static analysis of the structures. In the method, the structure is divided into substructures thus the number of unknowns that need to be worked out is reduced due to the transformation process. The static analysis of the structures can be performed easily and speedily by the proposed method. At the end of the study examples are presented for ensuring the agreement between the proposed method and classic Finite Element Method.

Perturbation Finite Element Transfer Matrix Method for Random Eigenvalue Problems of Uncertain Structures

The rapid computation of random eigenvalue problems of uncertain structures is the key point in structural dynamics, and it is prerequisite to the efficient dynamic analysis and optimal design of structures. In this paper, by combining finite element-transfer matrix method (FE-TMM) with perturbation method, a new method named as perturbation FE-TMM is presented for random eigenvalue problems of uncertain structures. By using the proposed method, the rapid computation of random eigenvalue problems of uncertain structures with complicated shapes and boundaries can be achieved, and the repeated eignvalues and characteristic vectors can be solved conveniently. Compared with stochastic finite element method, this method has the low memory requirement, high computational efficiency and high computational stability. It has more advantages for dynamic design of uncertain structures. Formulations as well as some numerical examples are given to validate the method.

Dynamic modeling and H ∞ independent modal space vibration control of laminate plates

In this paper, combining the transfer matrix method and the finite element method, the modified finite element transfer matrix method is presented for high efficient dynamic modeling of laminated plates. Then, by constructing the modal filter and the disturbance force observer, and using the feedback and feedforward approaches, the H ∞ independent modal space control strategy is designed for active vibration control of laminate plates subjected to arbitrary, immeasurable disturbance forces. Compared with ordinary dynamic modeling and control methods of laminated plate structures, the proposed method has the low memory requirement, high computational efficiency and robust control performance. Formulations as well as some numerical examples are given to validate the method and the control performance.

Modified Finite Element Transfer Matrix Method for Eigenvalue Problem of Flexible Structures

The speedy computation of eigenvalue problems is the key point in structure dynamics. In this paper, by combining transfer matrix method and finite element method, the modified finite element-transfer matrix method and its algorithm for eigenvalue problems are presented. By using this method, the speedy computation of eigenvalue problem of flexible structures can be realized, and the repeated eignvalue problem can be solved simply and conveniently. This method has the low order of system matrix, high computational efficiency, and stability. Formulations of this method, as well as some numerical examples, are given to validate the method.

Development of high speed computation algorithm for transient analysis of rectangular plates

A new transient analysis method for a rectangular plate structure comprised of a large number of plate elements was developed in order to significantly reduce computational time and memory. This algorithm was derived from the combination of the transfer technique of the transfer mass coefficient method, the modeling technique of the finite element method, and the numerical integration technique of Newmark’s method. In this paper, the algorithm for the transient analysis of a rectangular plate structure is formulated by the proposed method. In order to verify the computational accuracy and efficiency of the proposed method, the results obtained by the proposed method were compared with those obtained by the finite element method and the finite element-transfer matrix method. The proposed method, the finite element-transfer mass coefficient method, could considerably reduce the computation time without the loss of accuracy, in spite of using small computation memory, by using the transfer rules successively.

Saint-Venant Decay Rates for the Rectangular Cross Section Rod

A finite element-transfer matrix procedure developed for determination of Saint-Venant decay rates of self-equilibrated loading at one end of a semi-infinite prismatic elastic rod of general cross section, which are the eigenvalues of a single repeating cell transfer matrix, is applied to the case of a rectangular cross section. First, a characteristic length of the rod is modelled within a finite element code; a superelement stiffness matrix relating force and displacement components at the master nodes at the ends of the length is then constructed, and its manipulation provides the transfer matrix, from which the eigenvalues and eigenvectors are determined. Over the range from plane stress to plane strain, which are the extremes of aspect ratio, there are always eigenmodes which decay slower than the generalized Papkovitch-Fadle modes, the latter being largely insensitive to aspect ratio. For compact cross sections, close to square, the slowest decay is for a mode having a distribution of axial displacement reminiscent of that associated with warping during torsion; for less compact cross sections, slowest decay is for a mode characterized by cross-sectional bending, caused by self-equilibrated twisting moment.

A stiffness equation transfer method for transient dynamic response analysis of structures

A stiffness equation transfer method is proposed for transient dynamic response analysis of structures under various excitations. This method is a development and refinement of the combined finite element-transfer matrix (FE-TM) method. In the present method, the transfer of state vectors from left to right in the FE-TM method is changed into the transfer of general stiffness equations of every section from left to right. This method has the advantages of reducing the order of the ordinary transfer equation systems and minimizing the propagation of round-off errors occurring in recursive multiplication of transfer and point matrices. Furthermore, the drawback that the number of degrees of freedom on the left boundary must be the same as that on the right boundary in the ordinary FE-TM method, has now been avoided. The Newmark generalized acceleration formulation for time discretization is employed for a solution of the time problem. At the end, numerical examples are presented to demonstrate the accuracy as well as the potential of the proposed method for transient dynamic response analysis of structures.

A stiffness equation transfer method for natural frequencies of structures

A stiffness equation transfer method is proposed for obtaining vibration frequencies of structures. This method is an extension of the finite element-transfer matrix (FE-TM) method. In the present method, the transfer of state vectors from left to right in the ordinary FE-TM method is changed into the transfer of stiffness equations of every section from left to right. This method reduces the propagation of round-off errors produced in the ordinary transfer matrix method. Furthermore, the drawback that the number of degrees of freedom on the left boundary must be the same as that on the right boundary in the ordinary FE-TM method, is now avoided. Besides, this method finds out the values of the frequency by Newton–Raphson iteration method, so no plotting of the value of the determinant versus assumed frequency is necessary. An IFETM—W program based on this method for use on an IBM PC586 microcomputer is developed. Finally, numerical examples are presented to demonstrate the accuracy as well as the potential of the proposed method for free vibration analysis of structures.

A combined finite element–stiffness equation transfer method for steady state vibration response analysis of structures

An extended finite element transfer matrix method, in combination with stiffness equation transfer, is applied to dynamic response analysis of the structures under periodic excitations. In the present method, the transfer of state vectors from left to right in a combined finite element-transfer matrix (FE-TM) method is changed into the transfer of general stiffness equations of every section from left to right. This method has the advantages of reducing the order of standard transfer equation systems, and minimizing the propagation of round-off errors occurring in recursive multiplication of transfer and point matrices. Furthermore, the drawback that in the ordinary FE-TM method, the number of degrees of freedom on the left boundary be the same on the right boundary, is now avoided. A FESET program based on this method using microcomputers is developed. Finally, numerical examples are presented to demonstrate the accuracy as well as the potential of the proposed method for steady state vibration response analysis of structures.