The non-dimensional frequencies of a steel-alumina FG beam for different transverse varying distributions and boundary conditions.
Abstract
Over the recent past, various numerical analysis techniques have been formulated and used to obtain approximate solutions for numerous engineering problems to aid predict the behaviour of systems accurately and efficiently. One such approach is the Wavelet Finite Element Method (WFEM) which involves combining the classical Finite Element Method (FEM) with wavelet analysis. The key desirable properties exhibited by some wavelet families, such as compact support, multiresolution analysis (MRA), smoothness, vanishing moments and the ‘two-scale’ relations, make the use of wavelets in WFEM advantageous, particularly in the analysis of problems with strong nonlinearities, singularities and material property variations present. The wavelet based finite elements (WFEs) of a rod and beam are formulated using the Daubechies and B-spline wavelet on the interval (BSWI) wavelet scaling functions as interpolating functions due to their desirable properties, thus making it possible to alter the local scale of the WFE without changing the initial model mesh. Specific benchmark cases are presented to exhibit and compare the performance of the WFEM with FEM in static, dynamic, eigenvalue and moving load transient response analysis for homogenous systems and functionally graded materials, where the material properties continuously vary spatially with respect to the constituent materials.
Keywords
- multiresolution
- wavelets
- wavelet finite element (WFE)
- eigenvalue analysis
- moving load problem
- functionally graded material (FGM)
1. Introduction
In the analysis of complex structural problems, it is often challenging to formulate and apply exact closed-form solutions, as the realistic nature of such engineering systems exhibits varying complexities, high gradients and strong irregularities, e.g., suddenly varying loading conditions, contrasting material composition or geometric variations. Based on the existing mathematical tools available, such systems may require certain assumptions and generalisations to be implemented in order to simplify the model, which may lead to inability to correctly describe the properties and behaviour of the system under described conditions. However, the preferred approach is to find an approximate numerical solution, whilst retaining these complexities as accurately as possible, to better describe and predict the behaviour of such systems. This has given rise to numerical methods such as the classical Finite Element Method which employs polynomial interpolating functions to obtain approximate solutions for various engineering problems. Although this numerical analysis technique has grown in popularity, its use to tackle problems with regions of the solution domain where the gradient of the field variables are expected to vary suddenly or fast, bring on difficulties in the analysis of a complex system [1]. In order to improve on the accuracy and better represent the system’s behaviour, higher order polynomial interpolating functions or finer meshes may be employed and this in turn significantly increases the computational costs; which is undesirable. Moreover, the resolution of the elements can only be analysed to a specific scale once the orders of the governing polynomial functions have been selected. Subsequently, overcoming these challenges has been the driving force in the formulation of other numerical approximation techniques such as the Wavelet Finite Element Method [1, 2, 3, 4, 5, 6].
The initial development of wavelet analysis came from separate efforts that led to the foundation of modern wavelet theory. Grossman and Morlet [7] used wavelet analysis as a tool for signal analysis of seismic data and are credited with the introduction of the term and methodology of wavelets as it is known today. Ingrid Daubechies is recognised for her major breakthrough and contribution by constructing a family of orthonormal wavelet with compact support known as the Daubechies wavelets [8]. Wavelet analysis was used mainly by mathematicians as a decomposition tool for data functions and operators and its application has vastly grown in various disciplines at an exponential rate e.g., medicine [9], finance [10] and astronomy [11]. Likewise, the range of wavelet families and bases available for selection has also increased and this is credited to the properties of wavelets that allow it to be tailored to suite numerous avenues for design manipulation to meet the necessary and specific requirements for its application. The properties of different wavelet families vary, and therefore the decision on which family is the ‘most adequate’, is paramount to its application. Nevertheless, the more general aspects of wavelets formulations make it an important and convenient tool for mathematical manipulation allowing for the decomposition of a function into a set of coefficients that are dependent on scale and location. The ‘two-scale’ relation gives rise to one of the most key features of wavelet theory, multiresolution analysis (MRA), which allows for the convenient transformation of wavelet basis functions between different resolution scales [8]. Furthermore, the compact support property of wavelets ensures that the wavelet basis functions are finitely bound (non-zero over a finite range). The vanishing moments of wavelets allow the basic functions of wavelets to represent polynomials and other complex functions.
These desirable properties of wavelets have led to the use of wavelet basis functions as interpolating functions, in contrast to conventional polynomial functions as used in classical FEM, in the formulation of the wavelet based finite element method. For example, MRA permits for specific WFEs to be selected and analysed locally at finer scales without altering the initial system model, thus improving the accuracy of the solution, particularly in areas with high gradients or singularities present. Furthermore, rapid convergence of the method and compact support lead to a reduction in computational costs since fewer elements are required to achieve acceptable levels of accuracy [4, 5]. Due to the adaptability of wavelets, different wavelet families are being developed and customised for specific problems. However, it must be noted that when selecting a particular wavelet basis function for WFEM, key requirements, such as compatibility, completeness and convergence, must be satisfied and should allow for the easy implementation and treatment of boundary conditions.
The Daubechies wavelet based finite element was first introduced to solve a 1D and 2D second order Neumann problem via the formulation of a tensor product finite element [2]. The Daubechies wavelet Galerkin finite element was then used to analyse the bending of plates and beams [12] giving rise to the formulation of a wavelet based beam finite element [6] and two dimensional Daubechies wavelet plate finite element [13] for static analysis. The Daubechies wavelet base finite element stiffness matrices and load vectors were presented by Chen et al. at multiresolution scale
In a bid to overcome the limitations presented by the Daubechies wavelet, further research has been carried out to identify other potential wavelet families that can be implemented in WFEM. Basic spline functions were initially used as interpolating functions for the free vibration analysis of frame structures [3]. Chui and Quak [20] constructed the semi-orthogonal B-spline Wavelet on the Interval, which has the desirable properties of multiresolution, compact support, explicit expressions, smoothness and symmetry. The BSWI was employed to construct the wavelet based C0 type plane elastomechanics element and Mindlin plate element [21] as well as truncated conical shell wavelet finite elements [22]. Xiang et al. [5] significantly contributed to the use of BSWI in WFEM by constructing the axial rod, beam (Timoshenko and Euler Bernoulli) and spatial bar WFEs with a multiresolution lifting scheme. Furthermore, this research was extended to the static and dynamic analysis of plates based on Kirchhoff plate theory using BSWI based wavelet finite elements [23, 24]. Xiang et al. [25] were able to illustrate that the shear-locking phenomenon of a rotating Rayleigh-Timoshenko shaft was significantly eliminated when the BSWI based WFEs were employed. Majority of the problems examined by this point were of static analysis and this led Musuva and Mares [26] to develop and implement the Daubechies and BSWI homogenous beam WFEs for the analysis of dynamic response and moving load problems. The vibration and dynamic response analysis was carried out for frame structures using the two wavelet families [27] and the WFEM was compared with an analytical wavelet approach using coiflets for the analysis of vehicle-bridge interaction for fast moving loads [28]. Furthermore, the Daubechies and BSWI wavelets were used to construct a functionally graded beam wavelet finite element under various moving load conditions [29, 30].
Other different wavelet families have been selected and employed in the formulation of the WFEM to solve a wide variety of structural analysis problems and research in this field is still ongoing. The trigonometric Hermite wavelet, which can be explicitly expressed, was used to construct beam [31] and thin plate WFEs [32] for static and free vibration analysis. The Hermite Cubic Spline Wavelet on the Interval (HCSWI), polynomial wavelets [33] and the second generation wavelets [34] are other wavelet based approaches that have been introduced and researched on. A more comprehensive synthesis and summary of wavelet based numerical methods for various engineering problems is presented in [35].
A generalised Wavelet based Finite Element Method framework is presented based on the BSWI and Daubechies wavelet families to derive rod and beam WFEs for homogenous and functionally graded materials for static and dynamic structural problems. A brief introduction of wavelet analysis is described in Section 2, with emphasis given to the Daubechies wavelets, BSWI, multiresolution and connection coefficients formulations. In Section 3, the wavelet based finite elements for a rod, Euler Bernoulli homogeneous beam and transversely varying functionally graded beam are presented. The evaluation of the element matrices and various load vectors, including the WFEM moving load formulation, are presented. A comparison on the performance of the Daubechies and BSWI WFEMs are highlighted via numerical examples for a variety of static and dynamic structural problems in Section 4 followed by conclusions.
2. Wavelet and multiresolution analysis
Wavelets are a class of basic functions that represent functions locally, both in space and time, and allow for the analysis of functions to be carried out at different resolutions (scales) [36]. The wavelet basis emanates from a set of wavelet coefficients associated with a particular location in time and different multiresolution scales. The scaling and wavelet functions stem from multiresolution analysis (MRA), which is a key and desirable property of wavelets, and refers to the simultaneous appearance of multiple scales in function decompositions in the Hilbert space
Therefore in principle, in order for multiresolution to occur, the closed subspaces
The orthogonal complement subspace
Provided Eq. (6) and the above mentioned properties are satisfied, the wavelet orthonormal basis for subspace
The orthogonal subspaces
where
2.1. Daubechies wavelet
Daubechies wavelets are compact supported orthonormal wavelets developed by Ingrid Daubechies and for order
The scaling and wavelet functions have the supports
The scaling and wavelet functions defined in Eqs. (10)–(13) satisfy the following properties [8]:
Certain wavelet families have no explicit formulation, as is the case with the Daubechies wavelets. Therefore, Eq. (10) gives rise to a system of equations that require a normalising equation obtained from Eq. (14) to evaluate the scaling functions. The Daubechies wavelet of order
The coefficients
A normalising condition is required to evaluate Eq. (20) which is obtained from the moments of the scaling functions.
2.2. Daubechies connection coefficients
As earlier mentioned, the Daubechies functions cannot be computed analytically and their derivatives are highly oscillatory, particularly at low wavelet orders and/or high order derivatives. Therefore, the integral of the products of the scaling functions and/or derivatives are computed as what is commonly known as connection coefficients [37]. There are two forms of connection coefficients that are of relevance to this study; the multiscale two-term connection coefficient
where
Differentiating Eq. (23)
Substituting Eq. (24) into Eq. (22) and applying the ‘two-scale’ relation of the characteristic function, the two-term connection coefficient can be expressed as:
where
where the square matrix
Defining the second form of the connection coefficient
Substituting Eq. (27) into (28)
However,
Thus
where
2.3. B-spline wavelets on the interval [0,1] (BSWI)
The BSWI are a family of wavelets that emanate from Basis splines functions (B-Splines) and the basic functions in subspace
with the knot sequence
and have support
The number of inner scaling functions present in the formulation of BSWI is determined by the scale
The basis
where
and the scaling function derivatives can be evaluated directly by differentiating Eq. (40).
3. The wavelet finite element method
3.1. Axial rod wavelet finite element
Assume each WFE is divided into equal segments,
contains the unknown wavelet coefficients
The matrix
The matrix
Suppose the axial rod is subjected to nodal point loads
where
The stiffness matrix of the rod element in wavelet space,
In order for one to obtain the stiffness matrix in physical space, the element properties and transformation matrix
The load vector containing the axial point loads of the WFE in physical space is obtained as:
and the equivalent nodal load vector for the distributed load
When applying the Daubechies wavelet family, the WFE has a total of
where
3.2. Euler Bernoulli beam wavelet finite element
According to Euler Bernoulli beam theory, it is assumed that the shear deformation effects are neglected because before and after bending occurs, the plane cross-sections remain plane and perpendicular to the axial centroidal axis of the beam. The beam WFE of length
Therefore, the DOFs present within the entire beam element can be represented as
From Eq. (54), the transverse displacement and rotation at any point of the beam element can be expressed as:
where
where
This gives rise to the beam WFE stiffness matrix in wavelet space
The vector
The transverse kinetic energy of the beam element is expressed as
where
The mass matrix in physical space of the Euler Bernoulli beam element,
The vectors containing the element concentrated point loads, bending moments and equivalent distributed loads in physical space respectively are subsequently evaluated as:
In various engineering problems, the loading conditions analysed vary in location and/or magnitude with respect to time, e.g., a train travelling over a track, and this is generally referred to as moving load problems. Assume a moving load of magnitude
Therefore, the element load vector in physical space is evaluated as
Assuming the moving load transverses to a new position
When applying the Daubechies wavelet family of order
Correspondingly, the connection coefficients of the form
3.3. Transversely varying functionally graded Euler Bernoulli beam wavelet finite element
Functionally graded materials are a recent evolution of composite materials where the material constituents, hence properties, vary continuously in the desired spatial directions. The need for such revolutionary materials arose to overcome limitations of conventional composite materials, for instance, desirable properties would diminished when applied to highly intense thermal environments or material debonding due to increased stress concentration at material interfaces [42]. In the formulation of the wavelet based functionally grade beam as presented in Figure 4(a), of height
As illustrated in Figure 4(b), the transverse variation of the effective material properties
The beam WFE has axial deformation
where the vector
Therefore, the DOFs present within the entire beam element can be represented as
and consequently
The wavelet transformation matrix
where
The kinetic energy of the functionally graded beam element,
The wavelet based physical space elemental mass matrix of the beam,
4. Numerical examples
where
where
The FEM and WFEM governing equation is summarised as:
The vector
The matrices
BSWI | 4.34462 | 4.1943 | 3.84903 | 3.65795 | 3.37139 | 3.29504 | 3.33251 | ||
FEM | 4.34463 | 4.19431 | 3.84912 | 3.65811 | 3.3715 | 3.2951 | 3.33258 | ||
D | 4.34462 | 4.1943 | 3.84903 | 3.65795 | 3.37139 | 3.29504 | 3.33251 | ||
BSWI | 4.34462 | 4.1943 | 3.84903 | 3.65795 | 3.37139 | 3.29504 | 3.33251 | ||
BSWI | 5.43022 | 5.24233 | 4.81079 | 4.57197 | 4.21381 | 4.11838 | 3.92681 | ||
FEM | 5.43024 | 5.24238 | 4.81112 | 4.57253 | 4.21419 | 4.11856 | 3.92682 | ||
D | 5.43023 | 5.24234 | 4.8108 | 4.57197 | 4.21382 | 4.11839 | 3.92681 | ||
BSWI | 5.43022 | 5.24233 | 4.81079 | 4.57197 | 4.21381 | 4.11839 | 3.92681 | ||
BSWI | 6.54131 | 6.31498 | 5.79514 | 5.50745 | 5.07601 | 4.96105 | 4.73028 | ||
FEM | 6.54137 | 6.31509 | 5.79585 | 5.50867 | 5.07685 | 4.96145 | 4.73028 | ||
D | 6.54132 | 6.31498 | 5.79514 | 5.50745 | 5.07601 | 4.96106 | 4.73028 | ||
BSWI | 6.54131 | 6.31498 | 5.79514 | 5.50745 | 5.07601 | 4.96105 | 4.73028 | ||
BSWI | 2.59318 | 2.50345 | 2.29737 | 2.18333 | 2.01229 | 1.96671 | 1.87523 | ||
FEM | 2.59318 | 2.50346 | 2.2974 | 2.18337 | 2.01232 | 1.96673 | 1.87523 | ||
D | 2.59318 | 2.50345 | 2.29737 | 2.18333 | 2.01229 | 1.96671 | 1.87523 | ||
BSWI | 2.59318 | 2.50345 | 2.29737 | 2.18333 | 2.01229 | 1.96671 | 1.87523 | ||
BSWI | 8.68871 | 8.38806 | 7.69754 | 7.31541 | 6.74237 | 6.58969 | 6.66461 | ||
FEM | 8.68894 | 8.38834 | 7.69844 | 7.31684 | 6.74338 | 6.59024 | 6.66537 | ||
D | 8.68968 | 8.389 | 7.6984 | 7.31623 | 6.74313 | 6.59043 | 6.66535 | ||
BSWI | 8.68871 | 8.38806 | 7.69754 | 7.31541 | 6.74238 | 6.58969 | 6.66461 | ||
BSWI | 9.77473 | 9.4365 | 8.65966 | 8.22977 | 7.58512 | 7.41335 | 7.06849 | ||
FEM | 9.77513 | 9.43702 | 8.66145 | 8.23263 | 7.58713 | 7.41442 | 7.06879 | ||
D | 9.7765 | 9.43821 | 8.66124 | 8.23127 | 7.5865 | 7.4147 | 7.06977 | ||
BSWI | 9.77473 | 9.4365 | 8.65967 | 8.22977 | 7.58512 | 7.41335 | 7.06849 | ||
BSWI | 10.8597 | 10.4839 | 9.62083 | 9.14322 | 8.42702 | 8.23619 | 7.85305 | ||
FEM | 10.8604 | 10.4848 | 9.62387 | 9.14808 | 8.43044 | 8.238 | 7.85355 | ||
D | 10.8636 | 10.4877 | 9.62433 | 9.14655 | 8.43009 | 8.23918 | 7.8559 | ||
BSWI | 10.8597 | 10.4839 | 9.62083 | 9.14322 | 8.42702 | 8.23619 | 7.85305 | ||
BSWI | 6.49133 | 6.26671 | 5.75083 | 5.46534 | 5.03722 | 4.92315 | 4.69413 | ||
FEM | 6.49138 | 6.2668 | 5.75131 | 5.46615 | 5.03778 | 4.92342 | 4.69417 | ||
D | 6.49134 | 6.26673 | 5.75084 | 5.46535 | 5.03723 | 4.92316 | 4.69415 | ||
BSWI | 6.49133 | 6.26671 | 5.75083 | 5.46534 | 5.03722 | 4.92315 | 4.69413 | ||
BSWI | 13.0317 | 12.5808 | 11.545 | 10.9719 | 10.1125 | 3.29504 | 3.33251 | ||
FEM | 13.0334 | 12.5826 | 11.5489 | 10.9774 | 10.1166 | 3.2951 | 3.33258 | ||
D | 13.0461 | 13.5947 | 11.5578 | 10.984 | 10.1237 | 3.29504 | 3.33251 | ||
BSWI | 13.0317 | 12.5808 | 11.545 | 10.9719 | 10.1125 | 3.29504 | 3.33251 | ||
BSWI | 14.1176 | 13.629 | 12.507 | 11.8861 | 10.9551 | 10.7071 | 10.209 | ||
FEM | 14.1201 | 13.6318 | 12.5131 | 11.895 | 10.9617 | 10.711 | 10.2108 | ||
D | 14.1444 | 13.655 | 12.5308 | 11.9088 | 10.9761 | 10.7275 | 10.2284 | ||
BSWI | 14.1176 | 13.629 | 12.507 | 11.8861 | 10.9551 | 10.7071 | 10.209 | ||
BSWI | 15.2034 | 14.6773 | 13.4689 | 12.8003 | 11.7977 | 11.5306 | 10.9942 | ||
FEM | 15.2071 | 14.6813 | 13.4779 | 12.8135 | 11.8074 | 11.5364 | 10.9968 | ||
D | 15.2662 | 14.7379 | 13.5247 | 12.8533 | 11.8466 | 11.5783 | 11.0396 | ||
BSWI | 15.2034 | 14.6773 | 13.469 | 12.8003 | 11.7978 | 11.5306 | 10.9942 | ||
BSWI | 10.8611 | 10.4853 | 9.62205 | 9.14437 | 8.42814 | 8.2373 | 7.85409 | ||
FEM | 10.8618 | 10.4861 | 9.62437 | 9.14797 | 8.4307 | 8.23872 | 7.85458 | ||
D | 10.8667 | 10.4907 | 9.62702 | 9.1491 | 8.43249 | 8.24155 | 7.85814 | ||
BSWI | 10.8611 | 10.4853 | 9.62206 | 9.14438 | 8.42815 | 8.2373 | 7.8541 |
Assume the same beam, with simply supported boundary conditions and length
where
Critical velocity | Max | |||||||
---|---|---|---|---|---|---|---|---|
Ref. [46] | FEM | D | BSWI | Ref. [46] | FEM | D | BSWI | |
0 | 252 | 252 | 252 | 252 | 0.9328 | 0.9322 | 0.9323 | 0.9322 |
0.1 | – | 235 | 235 | 235 | – | 0.9863 | 0.9864 | 0.9863 |
0.2 | 222 | 222 | 222 | 222 | 1.0344 | 1.0340 | 1.0340 | 1.0340 |
0.5 | 198 | 198 | 198 | 198 | 1.1444 | 1.1435 | 1.1437 | 1.1436 |
1 | 179 | 178 | 178 | 178 | 1.2503 | 1.2491 | 1.2495 | 1.2493 |
2 | 164 | 164 | 164 | 164 | 1.3376 | 1.3363 | 1.3368 | 1.3365 |
3 | – | 157 | 158 | 158 | – | 1.3747 | 1.3751 | 1.3748 |
5 | – | 151 | 151 | 152 | – | 1.4217 | 1.422 | 1.4218 |
7 | – | 148 | 148 | 148 | – | 1.4567 | 1.4570 | 1.4568 |
10 | – | 145 | 145 | 145 | – | 1.4974 | 1.4976 | 1.4974 |
104 | 132 | 132 | 132 | 132 | – | 1.7308 | 1.7309 | 1.7308 |
5. Conclusions
A generalised formulation framework for the construction of an axial rod, Euler Bernoulli beam and functionally graded two-dimensional wavelet based finite elements is presented. The Daubechies and BSWI families are selected due to their desirable properties, particularly compact support, ‘two-scale’ relation and multiresolution. It is illustrated via a set of numerical examples that the WFEMs perform exceptionally well when compared to conventional
References
- 1.
Musuva M. The multiscale wavelet finite element method for structural dynamics [thesis]. London: Brunel University; 2015 - 2.
Ko J, Kurdila AJ, Pilant M. A class of wavelet-based finite element methods for computational mechanics. In: Proceedings of the 35th Structures, Structural Dynamics and Materials Conference, South Carolina; 1994 - 3.
Chen W-H, Wu C-W. A splines wavelets element method for frame structures vibration. Computational Mechanics. 1995; 16 (1):11-21 - 4.
Chen X, He Z, Xiang J, Li B. A dynamic multiscale lifting computation method using Daubechies wavelet. Journal of Computational and Applied Mathematics. 2006; 188 (2):228-245 - 5.
Xiang JW, Chen XF, He ZJ, Dong HB. The construction of 1D wavelet finite elements for structural analysis. Computational Mechanics. 2007; 40 (2):325-338 - 6.
Ma J, Xue J, Yang S, He Z. A study of the construction and application of Daubechies wavelet-based beam element. Finite Elements in Analysis and Design. 2003; 39 (10):965-975 - 7.
Grossmann A, Morlet J. Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM Journal of Mathematical Analysis. 1984; 15 (4):723-736 - 8.
Daubechies I. Orthonormal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics. 1988; 41 (7):909-996 - 9.
Rabbani H, Nezafat R, Gazor S. Wavelet-domain medical image denoising using bivariate Laplacian mixture model. IEEE Transactions on Biomedical Engineering. 2009; 56 (12):2826-2837 - 10.
Ramsey JB. The contribution of wavelets to the analysis of economics and financial data. Philosophical Transactions of the Royal Society. 1999; 357 (1760):2593-2606 - 11.
Starck J-L, Bobin J. Astronomical data analysis and sparity: From wavelets to compressed sensing. Proceedings of the IEEE. 2010; 98 (6):1021-1030 - 12.
Zhou Y, Wang J, Zheng X. Applications of wavelet Galerkin FEM to bending of beams and plate structures. Applied Mathematics and Mechanics. 1998; 19 (8):745-755 - 13.
Chen X, Yang S, Ma J, He Z. The construction of wavelet finite element and its applications. Finite Elements in Analysis and Design. 2004; 40 (5–6):541-554 - 14.
Chen M-Q, Hwang C, Shih Y-P. The computation of wavelet-Galerkin approximation on a bounded interval. Journal for Numerical Methods in Engineering. 1996; 39 (17):2921-2944 - 15.
Diaz LA, Martin MT, Vampa V. Daubechies wavelet beam and plate finite elements. Finite Elements in Analysis and Design. 2009; 45 (3):200-209 - 16.
Diaz LA, Vampa V, Martin MT. The construction of plate finite elements using wavelet basis functions. Revista Investigacion Operacional. 2009; 30 (3):193-204 - 17.
Wang YM, Chen XF, He ZJ. Daubechies wavelet finite element method and genetic algorithm for detection of pipe crack. Nondestructive Testing and Evaluation. 2011; 26 (1):87-99 - 18.
Mitra M, Gopalakrishnan S. Spectral formulated wavelet finite element for wave propagation and impact force identification in connected 1D waveguides. International Journal of Solids and Structures. 2005; 42 (16–17):4695-4721 - 19.
Zhao B, Wang K. The application of the wavelet finite element method on the temperature field and thermal stress analysis of the petroleum tank. Journal of Information and Computational Science. 2011; 8 (7):1103-1111 - 20.
Chui CK, Quak E. Wavelets on a bound interval. Numerical Methods of Approximation Theory. 1992; 9 :53-57 - 21.
Xiang J, Chen X, He Y, He Z. The construction of plane elastomechanics and Mindlin plate elements of B-spline wavelet on the interval. Finite Elements in Analysis and Design. 2006; 42 (14–15):1269-1280 - 22.
Xiang J, He Z, Chen X. The construction of wavelet-based truncated conical shell element using B-spline wavelet on the interval. Acta Mechanica Solida Sinica. 2006; 19 (4):316-326 - 23.
Xiang J, Chen X, He Y, He Z. Static and vibration analysis of thin plates by using finite element method of B-spline wavelet on the interval. Structural Engineering and Mechanics. 2007; 25 (5):613-629 - 24.
Xiang J, Chen X, He Z, Zhang Y. A new wavelet-based thin plate element using B-spline wavelet on the interval. Computational Mechanics. 2008; 41 (2):243-255 - 25.
Xiang J, Chen D, Chen X, He Z. A novel wavelet-based finite element method for the analysis of rotor-bearing systems. Finite Element in Analysis and Design. 2009; 45 (12):908-916 - 26.
Musuva M, Mares C. The wavelet finite element analysis of a beam subject to a moving load. In: Proceedings of the First International Conference on Railway Technology: Research, Development and Maintainance, Stirlingshire, UK; 2012 - 27.
Musuva M, Mares C. Vibration analysis of frame structures using wavelet finite elements. In: Journal of Physics: Conference Series 382, Bristol; 2012 - 28.
Musuva M, Koziol P, Mares C, Neves M. The analysis of beams subject to moving loads using: Coiflets, the wavelet finite element method and the finite element method. In: Proceedings of the Second International Conference on Railway Technology: Research, Development and Maintenance, Stirlingshire, UK; 2014 - 29.
Musuva M, Mares C. The dynamic analysis of functionally graded materials (FGM) using the wavelet finite element method (WFEM). In: Proceedings of the 26th International Conference on Noise and Vibration Engineering, Leuven, Belgium; 2014 - 30.
Musuva M, Mares C. The wavelet finite element method in the dynamic analysis of a functionally graded beam resting on a viscoelastic foundation subjected to a moving load. European Journal of Computational Mechanics. 2015; 24 (5):171-209 - 31.
He W-Y, Ren W-X. Finite element analysis of beam structures based on trigonometric wavelet. Finite Elements in Analysis and Design. 2012; 51 :59-66 - 32.
He W-Y, Ren W-X. Trigonometric wavelet-based method for elastic thin plate analysis. Applied Mathematical Modelling. 2013; 37 (4):1607-1617 - 33.
Castro LMS. Polynomial wavelets in hybrid-mixed stress finite element models. International Journal for Numerical Methods in Biomedical Engineering. 2010; 26 (10):1293-1312 - 34.
Wang YM, Chen XF, He ZJ. An adaptive inverse iteration algorithm using interpolating multiwavelets for structural eigenvalue problems. Mechanical Systems and Signal Processing. 2011; 25 (2):591-600 - 35.
Li B, Chen X. Wavelet-based numerical analysis: A review and classification. Finite Elements in Analysis and Design. 2014; 81 :14-31 - 36.
Chui CK. An Introduction to Wavelets. 1st ed. London: Academic Press Limited; 1992 - 37.
Latto A, Resnikoff HL, Tenenbaum E. The evaluation of connection coefficients of compactly supported wavelets. In: Proceedings of the French-USA Workshop on Wavelets and Turbulence, New York; 1991 - 38.
Quak E, Weyrich N. Decomposition and reconstruction algorithms for spline wavelets on a bounded interval. Applied and Computational Harmonic Analysis. 1994; 1 (3):217-231 - 39.
Goswami JC, Chan AK, Chui CK. On solving first-kind integral equations using wavelets on a bounded interval. IEEE Transactions on Antennas and Propagation. 1995; 43 (6):614-622 - 40.
Logan DL. In: James H, editor. A First Course in the Finite Element Method. 4th ed. Toronto: Nelson; 2006 - 41.
Fryba L. In: Stott E, editor. Vibration of Solids and Structures under Moving Loads. 3rd ed. London: Thomas Telford Ltd.; 1999 - 42.
Carrera E, Giunta G, Petrolo M. Beam Structures: Classical and Advanced Theories. Wiley; 2011 - 43.
Wakashima K, Hirano T, Niino M. Space applications of advanced structural materials. In: European Space Agency: Proceedings of an International Symposium, Paris, France; 1990 - 44.
Bondi B, Caddemi S. Closed form solutions of Euler-Bernoulli beams with singularities. International Journal of Solids and Structures. 2005; 42 (9–10):3027-3044 - 45.
Bathe KJ. Finite Element Procedures. 1st ed. New Jersey: Prentice Hall; 1996 - 46.
Simsek M, Kocaturk T. Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Composite Structures. 2009; 90 (4):465-473