## 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 *j* = 0 [14] and different multiresolution scales [4]. The Daubechies plate finite element was developed by Diaz et al. for the static analysis of plates based on Mindlin-Reissner plate theory [15], where shear deformation is taken into consideration through the thickness of the plate, and compared it with Kirchhoff plate theory formulations [16]. This wavelet family has also been used in the analysis of many other structural problems, including formulation of the Rayleigh-Euler and Rayleigh-Timoshenko beam elements [17], the wavelet based spectral finite element to study elastic wave propagation in 1-D connected waveguides [18] and also to investigate the thermal stress distribution along the vertical direction of the tank wall [19]. Overall, the wavelet family performed decently in providing accurate solutions for the various structural analysis problems tackled. However, the Daubechies wavelet lacks an explicit expression for the wavelet and scaling functions and possesses unusual smoothness characteristics, particularly for lower orders, making it challenging to evaluate the numerical integrals necessary for the formulation of the element matrices and load vectors. The evaluation of the connection coefficients is therefore necessary for the formulation of these element matrices and vectors.

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 C^{0} 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 *V _{j},* which is represented mathematically as [36]:

Therefore in principle, in order for multiresolution to occur, the closed subspaces

The orthogonal complement subspace *V _{j}* and subsequent subspace

*V*

_{0}. The orthonormal basis for the next space

Provided Eq. (6) and the above mentioned properties are satisfied, the wavelet orthonormal basis for subspace *j* is

The orthogonal subspaces *j* in the subspaces

where

### 2.1. Daubechies wavelet

Daubechies wavelets are compact supported orthonormal wavelets developed by Ingrid Daubechies and for order *L*, the scaling and wavelet functions are described by the ‘two-scale’ relation [8]:

The scaling and wavelet functions have the supports *V _{j}* and

*W*are defined as:

_{j}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 *L* has *j* can represent a polynomial of order

The coefficients *V _{j}*. The derivatives of the Daubechies wavelet scaling functions are evaluated by differentiating the refinement Eq. (10)

*m*times, and are obtained as [12]:

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 *a* and *b* are the orders of the scaling function at multiresolution *j*, while the values *a* and *b* at different multiresolution scales *j*. From the ‘two-scale’ relation presented in Eq. (10),

Differentiating Eq. (23) *m* times

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 *m* and scale

with the knot sequence

*m*^{th} divided difference of the truncated power function *t*. The general B-splines take the form

and have support *m*-tuple knots at 0 and 1, as expressed in Eq. (33). The knots at 0 ad 1 coalesce and form multiple knots for BSWI while the internal knots are simple hence smoothness is unaffected. For the knot sequence on [0,1],

The number of inner scaling functions present in the formulation of BSWI is determined by the scale *j*. There must be at least one inner scaling function on the interval [0,1] and this gives rise to the minimum value of *j* necessary to ensure this condition is met and is defined as

The basis *m*^{th} cardinal B-splines, *j* [38]:

where *m* times. The corresponding B-wavelet with support

*m*^{th} derivative for the B-spline of order 2*m* and scale

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, *n _{s}*, connected by

**Figure 1**, with axial deformation

**Figure 2(a)**, where

*i*corresponding to coordinate position

*x*. The nodal natural coordinates is

_{i}*ξ*≤ 1, 1 ≤

_{i}*i*≤

*r*). The Daubechies and BSWI scaling functions

*z*at multiresolution scale

*j*, the axial deformation

contains the unknown wavelet coefficients

The matrix

The matrix *r* and *w* denoting rod and wavelet respectively. The wavelet based axial rod shape functions can be evaluated as

Suppose the axial rod is subjected to nodal point loads

where *E* is the Young’s modulus, *A* is the cross-sectional area and *l* is the length of the rod. Therefore, given the relation highlighted in Eq. (44), the axial stain energy

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 *m* depends on the order of the function

### 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 *L _{e}*, is divided into

*n*equally spaced elemental segments connected by

_{s}*r*elemental nodes at coordinate values

**Figure 1**. The WFE has the transverse displacement

*v*and rotation

*θ*taken into account, with corresponding transverse forces

*n*as illustrated in

**Figure 2(b)**. Therefore, there are

*ξ*≤ 1, 1 ≤

_{i}*i*≤

*r*). The deflection and rotation at any point of the wavelet based beam finite element can be approximated by applying the wavelet scaling functions

*z*at multiresolution scale

*j*as interpolating functions.

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 *E* is the Young’s modulus, *I* is the moment of inertia and *l* is the length of the beam. The strain energy

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 *A* is the cross-sectional area of the beam. Applying the scaling functions to approximate the displacements within the beam, the kinetic energy becomes

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:

(63) |

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 *P* travels across a beam element, as illustrated in **Figure 3**, from the left at a constant speed of *c* m·s^{−1} and is represented by the function *t*. The potential work of the load at this instant at position

Therefore, the element load vector in physical space is evaluated as

Assuming the moving load transverses to a new position *t*. When the moving load is acting on a new WFE, the scaling functions corresponding to the WFE subjected to the moving load are used to obtain the load vector for that particular element.

When applying the Daubechies wavelet family of order *L* at multiresolution *j*, the total DOFs within a single element is

Correspondingly, the connection coefficients of the form *m* is based on the load function *m* and at scale *j*, there are

### 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 *h*, length *l* and width *b,* the material distribution is modelled based on the power law of transverse gradation [43]

As illustrated in **Figure 4(b)**, the transverse variation of the effective material properties *P*(*y*) (Young’s modulus) can be infinitely altered via the non-negative volume fraction power law exponent, *n*.

The beam WFE has axial deformation **Figure 4(c)**. The wavelet scaling functions are implemented as interpolating functions and the axial deformation, deflection and rotation at any point of the beam element are described by Eqs. (42) and (53) respectively. However, in order to ensure that the defined DOFs are positioned correctly, the layout of the element determines the order of scaling functions selected. In this case, the order of the scaling functions selected to approximate the axial displacement is *s*, present in the functionally graded beam element is

(71) |

where the vector

(72) |

Therefore, the DOFs present within the entire beam element can be represented as

and consequently

(74) |

The wavelet transformation matrix

(75) |

where

(76) |

(77) |

The kinetic energy of the functionally graded beam element,

(78) |

(79) |

The wavelet based physical space elemental mass matrix of the beam,

## 4. Numerical examples

**Example 1:** A uniform axial cantilever rod (free-fixed) subjected to linear varying load *l.* The exact solution for displacement at a particular point *x* can be obtained by solving *h*-FEM and *p-*FEM formulations. The governing equation of the system for FEM and WFEM is

where *l* and the rate of convergence of the different approaches is compared in **Figure 5**. The plot shows the absolute relative error of the axial deformation and corresponding number of DOFs. The FEM (*h*-FEM) solution involves increasing the number of elements, *p*-FEM involves increasing the order of the polynomials (one element only) and both Daubechies and BSWI WFEMs have the order and/or multiresolution scale *j* increased. The results show that although the rates of convergence of all the methods are similar, the WFEM approaches have a slightly improved rate with only one element employed.

**Example 2:** A simply supported two-stepped beam of length 2*l* has non-uniform flexural stiffness represented by the unequal cross sections; the bending stiffness of the right and left half is given as *q*(*x*) = 1. The flexural stiffness function is expressed as [44]:

where

The FEM and WFEM governing equation is summarised as:

The vector *h*-FEM (FEM-8; 8 elements), *p*-FEM of order 9 (*p-*FEM-9; 2 elements), Daubechies WFEM of order _{1}; 2 elements) and the BSWI WFEM of order _{3}; 2 elements) are selected for comparison with the exact solution governed by Eq. (83). Each approach has a total of 18 DOFs within the beam. The deflection and rotation across the beam is presented in **Figure 6(a)** and **(b)** respectively. The percentage errors of the deflections are compared for the different approaches and presented in **Figure 6(c)**. All numerical approaches describe the deflection and rotation across the beams very accurately. However, given that both the Daubechies and BSWI WFEM deflection solutions have a maximum error of 1.28% in comparison to 3.82% from the *h-*FEM and *p*-FEM approaches, the WFEMs exhibit better convergence to the exact solution. Furthermore, improved accuracy is attained with fewer elements implemented than the *h-*FEM and *p*-FEM and this results in reduced computational time.

**Example 3:** A steel-alumina functionally graded beam of length *l* and uniform cross-sectional area ^{2} (height ^{−3} and ^{−3} respectively (*E* and *ρ* denote the Young’s modulus and density respectively. The slenderness ratio for the beam is *n* in Eq. (70). The free vibration of the functionally graded beam is governed by [45]

The matrices *i*^{th} non-dimensional frequency *;* 37 DOFs); one BSWI (*h-*FEM elements (39 DOFs). The results of the first 3 non-dimensional natural frequencies of the beam are presented in **Table 1** for different boundary conditions and material distributions. It is observed that all approaches give highly accurate results with respect to the reference (BSWI*5 _{5}*), particularly for the fundamental frequencies. Furthermore, the BSWI WFEM solution exhibits better levels of accuracy than the Daubechies WFEM and

*h*-FEM solutions for the higher frequencies. Both WFEM solutions achieve high levels of accuracy with the described layout of having the rotation DOFs present at elemental and nodes and using fewer elements that the

*h*-FEM approach.

Assume the same beam, with simply supported boundary conditions and length *c* m·s^{−1}. The behaviour of the beam is described using Euler Bernoulli beam theory and is assumed to be undamped. The governing equation describing the dynamic behaviour of the system is given by [45]:

where *t*. *P* at the centre. The maximum normalised deflection mid-span of the beam is analysed over a moving load velocity range ^{−1} at increments of 1 m·s^{−1} to identify the critical velocity for the different variations of the constituent materials as illustrated in **Figure 7**. The results present are obtained from the BSWI (2 element; *h*-FEM (12 elements; 39 DOFs) and Daubechies (2 elements; *;* 37 DOFs) WFEM solution gives similar results. The values of the critical moving load velocity and corresponding maximum non-dimensional displacement are presented in **Table 2** for the different values of *n* for all approaches. The results are compared with those presented in [46]. Both the Daubechies and BSWI WFE M solutions very accurately yield the correct values.

n | Critical velocity c m∙s^{−1} | Max | ||||||
---|---|---|---|---|---|---|---|---|

Ref. [46] | FEM | D12_{0} | BSWI4_{3} | Ref. [46] | FEM | D12_{0} | BSWI4_{3} | |

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 |

10^{4} | 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 *h*-FEM and *p-*FEM where high levels of accuracy are achieved with fewer elements required and the approaches converge more rapidly to the exact solution. Furthermore, the methods are able to accurately describe the behaviour of static and dynamic systems with singularities, variation in material properties and loading conditions present. This exhibits the vast potential of the method in the analysis of more complicated systems and the ability to alter the multiresolution scales without affecting the original mesh allows effective and efficient avenues solution accuracy improvement.