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This article introduces a groundbreaking finite element method that leverages trigonometric shape functions with hidden nodes, specifically designed for dynamic simulations of rods and beams. The study delves into the numerical formulation of these non-conventional finite elements, showcasing their application in modal and transient dynamic analyses. Through rigorous case studies, the article demonstrates the superior accuracy and computational efficiency of the proposed method compared to conventional 3-node finite elements. The research highlights the method's ability to predict natural frequencies with greater precision, particularly in the higher frequency range, which is crucial for explicit time integration schemes. Additionally, the article explores the convergence characteristics and computational performance of the new elements, revealing significant speed-ups in transient dynamic simulations. The findings underscore the potential of trigonometric-based finite elements to enhance the accuracy and efficiency of structural dynamic analyses, making them a promising alternative to traditional polynomial-based methods.
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Abstract
This work presents a new type of non-conventional finite elements (FE) that utilize trigonometric-based shape functions. The selection of the shape functions is inspired by the analytical expression of structural modeshapes. The proposed element consists of two “regular” nodes and a middle “hidden” node that basically enriches the local approximation and leads to the partition-of-unity property. Two different element types are constructed: a rod element with axial degrees of freedom and a Timoshenko beam element with two degrees of freedom: vertical displacement and rotation. Both the proposed elements are tested against conventional 3-node FE in free vibration and transient dynamic simulations of isotropic rod and beam structures. Numerical results show that the proposed trigonometric FE yield more accurate estimations of natural frequencies than the traditional 3-node FE. Also, the maximum natural frequency of each case is not only more accurate but also has smaller numerical value. This leads to the selection of larger time steps when employing explicit time integration, resulting in lower computing times. Finally, the presented elements evince higher convergence rates than the conventional 3-node FE in wave propagation simulations of rods and beams, further increasing the proposed method’s efficiency. This is explicitly quantified, since the proposed FE appears to be twice as fast as the conventional 3-node FE, in obtaining a transient wave response with the same level of accuracy.
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1 Introduction
The finite element method (FEM) is a powerful computational tool for the numerical analysis of physical problems [1]. Various types of finite elements (FE) have been created and investigated for several types of structural problems, such as static, steady-state dynamic and transient dynamic among others. One-dimensional elements for beams have already been studied and developed both using classical beam theory (Euler-Bernoulli) [2] and high-order shear beam theories such as Timoshenko–Ehrenfest beam theory [3] and more complex layerwise theories [4‐6]. In most cases, simple low-order polynomial functions are used for the discretization of the physical domain because of their easy formulation and broad range of applicability. Finite elements that utilize shape functions with order above two are mostly avoided due to the Runge phenomenon [7]. To overcome this limitation, several types of spectral FE have been developed, utilizing non-uniform grid of nodes so as to surpass the Runge phenomenon even with high-order polynomial functions [8‐14].
More elaborate shape functions such as B-splines and their variations have been utilized, forming the Isogeometric analysis (IGA) [15, 16]. The IGA is a very powerful option since the geometric error is practically eliminated. Specifically, IGA was employed for the static analysis of Euler-Bernoulli beams, showcasing excellent results [17]. In general, spline-based finite elements were investigated even earlier by Leung and Au [18] in free vibration analyses of beams and plates. They were greatly motivated by the spline-finite-strip method, developed by Cheung et al. [19‐21]. Another variation of the IGA is the T-spline FE or T-spline isogeometric analysis that is excessively studied in linear and nonlinear static simulations of plate and shell structures [22‐24]. Moreover, Kuhn et al. [25] developed finite elements using quintic and biquintic B-splines, for the static solution of beam and plate clamped structures. In a different direction, wavelet functions have been also utilized as basis functions for numerical methods due to their remarkable properties [26]. Several wavelet families such as the Haar wavelet [27], spline wavelets [28, 29], Deslauriers-Dubuc interpolets [30, 31], Coiflets [32, 33] and Daubechies wavelets [34‐36] were utilized for the static and dynamic simulation of beam structures. Despite the high efficiency and accuracy of those methods, their formulation and utilization is rather complex. In consequence, simpler functions such as trigonometric, exponential or logarithmic functions deserve further examination in their capacity as shape functions for computational methods.
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Exponential functions are mostly utilized as shape functions in special cases, such the approximation of damage in phase field models [37] and the modeling of shear boundary layers in bearings [38]. In addition, logarithmic functions were employed by Schröppel and Wackerfuß [39], creating the logarithmic finite element method, also called LogFE, that manifested very good results in one-dimensional static problems of beam structures. They also expanded their method for large displacements rotations of Timoshenko beams [40]. In a different direction, using trigonometric functions as a base for shape functions is a popular approach specifically when problems involve curved beams or shell-shaped structures due to the usage of different (cylindrical or spherical) coordinate system [41]. Moreover, trigonometric-based FEM do not exhibit shear or membrane locking [42, 43] making them very promising as an alternative of other options such as reduced integration. Hansen et al. [43] developed a Reissner–Mindlin shell element with trigonometric-based shape functions. Their results showed good convergence and accuracy compared with classical shell formulations and were tested on several case studies for various cylindrical geometries. Heppler et al. [42] extended the usage of these shape functions in Timoshenko curved beams. The results were competitive with classical 3-node FEM elements with polynomial base functions both in terms of convergence and accuracy. Additionally the formulated elements do not suffer from shear and membrane locking. Hashemi et al. [44] used trigonometric shape functions for the free vibration analysis of spinning beams using Euler-Bernoulli beam theory. Their method exhibited very good accuracy and higher convergence rates compared to conventional finite element models. The trigonometric-based shape functions used in all the above-mentioned works follow a Lagrange polynomial-type formulation, but they utilize sine functions instead of simple binomials.
In this study, a new type of non-conventional finite element is developed, employing trigonometric-based shape functions for the approximation of the state variables. A combination of both sine and cosine functions is utilized in order to increase the convexity of the shape functions. The rationale behind the present formulation is inspired by the fact that in free vibration, steady-state dynamics and transient dynamics, the deformed structural shape generally follows trigonometric patterns. The proposed element has three nodes, two regular outermost nodes and a hidden middle node. The middle node is termed as hidden, since the second shape function is utilized as an enrichment function and does not fully satisfy the classical continuity conditions. Numerical case studies of modal and transient dynamic simulations in rods and beam evince that the proposed trigonometric FE outperform the 3-node conventional FE both in terms of accuracy and computational efficiency.
The rest of the paper is organized as follows. In Section 2, the numerical formulation of the proposed non-conventional FE is described. The utilized shape functions are presented and the construction of the stiffness and mass matrices for rod and Timoshenko beam elements is introduced. In Section 3, numerical case studies of modal analyses and transient dynamic simulations take place, for both rod and beam structures. The transient dynamic simulations are implemented using explicit time integration. Furthermore, convergence studies are performed, displaying the computational efficiency of the proposed method and finally, CPU time comparisons manifest the superiority of the proposed FE compared to conventional 3-node FE. At last, in Section 5, concluding remarks and future work ideas are outlined.
2 Numerical formulation
2.1 Shape functions
The proposed method is intended to be utilized in dynamic problems, since static simulations are simpler and so the traditional finite elements perform adequately in such problems. For this reason, trigonometric functions are employed, because of their resemblance with modeshapes and dynamic structural deformation, in general. The shape functions chosen as basis for the two outermost nodes of the proposed non-conventional FE are described with trigonometric terms and expressed in general form in the equations below:
where \(\alpha , \ \beta , \ \gamma \) are variables to be determined based on the classic continuity conditions and the term \(\pi /2\) ensures the symmetry of the shape functions inside the element domain with respect to the middle node. Specifically, in order for \(N_I\) and \(N_{II}\) to satisfy the continuity conditions, the shape functions are given in dimensionless form as:
Those shape functions are calculated in a local coordinate system in the domain \(\xi \)\(\in \) [-1,1] with a linear transformation from the physical domain [0, \(L_e\)], such that \(x = \frac{L_e}{2}(1+\xi )\). However, those two shape functions do not satisfy the partition-of-unity property. That is why an enrichment shape function (\(N_e\)) is also involved, inspired by the generalized finite element method (GFEM) [45], in which the partition of unity is achieved by multiplying the "standard" shape functions with local enrichment functions. Along with this enrichment functions there also exists the respective enrichment node, that lies at the middle of the elemental length and is termed as hidden node. This enrichment shape function is given as:
For convenience purposes, the shape functions are named as \(N_1, \ N_2, \ N_3\), hereafter. In particular, the shape functions are finally expressed as:
The functions \( N_1 \) and \( N_3 \) were created by satisfying the general continuity conditions of the FE basis functions. However, the shape function of the second node, \(N_2\), does not satisfy the continuity condition of being equal to 1 for node 2 (\(x=L_e/2, \ \xi =0\)). The shape function \(N_2\) is used to ensure that the partition of unity (\(N_1 + N_2 + N_3 = 1\)) is satisfied in the whole domain of the element. The shape functions and their corresponding derivatives (\(R_i=dN_i/dx\)) are presented in Fig. 1. As already mentioned, the selection of the specific basis functions was inspired by the modeshapes of the dynamic response of structures that in general involve trigonometric terms. Also the derivatives of \(N_1\), \(N_2\) and \(N_3\) are smooth, since they are derived from trigonometric functions, and their summation leads to a constant zero value (\(R_1+R_2+R_3=0\)), which is desired, since zero strain is expected when the three displacements of an element have equal values (Fig. 1b).
Fig. 1
a Elemental shape functions and b their derivatives, inside the normalized element domain. The nodes of the element are presented with circles
The classic rod 3-node finite element has three degrees of freedom (DOF) in the axial direction (x-direction) denoted as \(u_1, u_2, u_3\). Also, the axial strain is given as \(\epsilon _x=du/dx=\sum _{i=1}^{3} R_i u_i\). Following the principle of virtual work, the general formulation of the elemental stiffness matrix terms for a 3-node finite element is expressed as:
$$\begin{aligned} {K}^{ij}_e = E A \int _{0}^{L_e}R_i(x)R_j(x)dx \quad i,j = 1,2,3 \end{aligned}$$
where \(R_{i,j}\) correspond to the derivatives of the shape functions, and \(E\), \(A\) are the elastic modulus and cross-sectional area of the rod. Additionally the elemental mass matrix terms are given by:
where \(\rho \) is the material’s density and \(N_{i,j}\) are the shape functions. The integrals on the Equations 6 and 8 are numerically calculated, and the full stiffness and mass matrices are presented in Appendix A.
2.3 Timoshenko beam element
For the Timoshenko beam element formulation, we use two DOFs in each node, the transverse displacement \(w_0^i\) and the rotation of the beam’s cross section \(\beta _x^i\), where \(i\) indicates the node number. The vector including the nodal degrees of freedom is termed as \(U^i=[w_0^i \ \beta _x^i]^T\). The generalized Hooke’s law is well known and takes the form for a general anisotropic beam as:
where \(Q_z, M_x\) are the shear force and the bending moment, and \(A_{55}\) and \(D_{11}\) are the shear and bending stiffness terms of the cross section. Also, the generalized strain vector is given as \(\varepsilon ^i=[k_x^i \ \varepsilon _{xz}^i]^T\). For an isotropic beam, \(A_{55}=\kappa GA\) and \(D_{11}=EI\), where \(\kappa \) is the shear correction factor that is considered equal to 1 for this work, G is the shear modulus and I is the second moment of inertia, \(k_x=\frac{d\beta _x}{dx}\) is the beam’s curvature and \(\varepsilon _{xz}=\beta _x+\frac{dw_0}{dx}\) is the beam’s shear strain. The selection of the shear correction factor being equal to 1 has no practical effect in the results of the paper since this value is also utilized in all models and analytical expressions used for comparing the proposed model results in the next sections. After all, changing the shear correction factor is just a minor input change in the utilized script or software. Following the Principle of Virtual Work and transforming the coordinate system as in Section 2, we get the final formulation of the elemental stiffness and mass submatrices:
where \(\mathbf {[N_i]_{2 \times 2}}=diag(N_i)_{2 \times 2}\), \(\rho _A=\rho A \) and \(\rho _D=\rho I \) for structures with homogeneous density through-the-thickness. It is highlighted that the generalized strain vector is given in terms of the nodal displacement degrees of freedom, as \(\varepsilon ^i=\mathbf {[R_i]}\cdot U^i\). For the implementation of Equations 10 and 12 one needs to calculate a series of integrals of the terms \(R_i(\xi ) \cdot R_j(\xi )\), \(N_i(\xi ) \cdot N_j(\xi )\) and \(R_i(\xi ) \cdot N_j(\xi )\). Those integrals are numerically calculated and presented in Appendix A.
3 Case studies
3.1 Free vibration analysis
The generalized eigenvalue problem for a conservative system with mass matrix \(\mathbf {[M]}\), stiffness matrix \(\mathbf {[K]}\) and generalized displacements \(\mathrm {\underline{u}}\) is given as:
where \( \mathbf {[K_{DYN}]}=(-\omega ^2 \mathbf {[M]} + \mathbf {[K]})\) is the well-known dynamic stiffness and \(\omega \) are the eigenfrequencies. So, in the next subsections, the two matrices \(\mathbf {[M]}\) and \(\mathbf {[K]}\) that construct the dynamic stiffness and the generalized displacement vector \(\mathrm {\underline{u}}\) refer to the rod or beam element, as described in Section 2.
3.1.1 Rod
A simple numerical case study was conducted to demonstrate the overall performance of the proposed element. The natural frequencies were calculated for a clamped-free isotropic rod with the geometrical and material characteristics that are presented in Table 1.
Table 1
Geometric and material characteristics for modal case studies
Property
Rod
Beam
Material Properties
\(L \ (m)\)
4
2
\(E=70\ GPa\)
\(b \ (m)\)
\(10 \times 10^{-3}\)
\(10 \times 10^{-2}\)
\(\rho = 2700\ kg/m^3\)
\(h \ (m)\)
\(10 \times 10^{-3}\)
\(10 \times 10^{-2}\)
\(v = 0.3\)
The results of the simulations are compared to classical 3-node rod finite elements that incorporate second-order polynomial shape functions. It is mentioned that in this paper, the terminology "classical," "conventional" or "traditional" 3-node FE, implies the Lagrangian \(C^0\) quadratic FE. The results are normalized with the analytical solution given as:
and shown in Fig. 2 in a combined plot. For both the trigonometric and traditional 3-node FE, 100 elements are utilized, so 200 eigenfrequencies are obtained and compared to the analytical equation (14). The eigenfrequencies are equal to the degrees of freedom of the bounded system, namely, the rod after imposing the clamped-free boundary conditions. It is clear that the proposed trigonometric FE outperform the classical 3-node FE for every mode. It should be highlighted that the proposed method results in a significantly smaller value of the spectral radius of the dynamic stiffness. That means that it yields smaller values for the structure’s highest natural frequency that determines the maximum timestep that should be used in explicit integration schemes (Courant–Friedrichs–Lewy condition or CFL condition [46]). In this way, the more accurate prediction of the highest natural frequency plays a pivotal role on the numerical efficiency of transient dynamic simulations, as demonstrated in subsection 3.2. More specifically, the proposed trigonometric FE yield more accurate results on higher frequencies, showing a 17% overestimation of the highest frequency compared to the classical 3-node FE that exhibits 24% overestimation.
At this point, a convergence analysis is performed for the estimation of the 20th natural frequency, following the procedure described in [47, 48]. The absolute error between the estimated and the analytical 20th natural frequency, as well as the convergence rates as described in [48], is presented in Table 2, for the classic 3-node FE and the proposed trigonometric FE. It is obvious that the proposed FE exhibit lower absolute errors in every discretization case, and the convergence rates of the two models are similar.
Fig. 2
Ratio of numerical to analytical natural frequencies (NF) for a clamped-free rod using conventional 3-node FE and the proposed trigonometric FE
Comparison of absolute errors and convergence rates for the conventional 3-node FE and proposed trigonometric FE, in the estimation of rod’s 20th natural frequency
Conventional 3-node FE
Proposed trigonometric FE
Mesh
Abs. Error
Conv. Rate
Abs. Error
Conv. Rate
20 el.
3218
–
2480
–
40 el.
289.5
3.474
217.9
3.509
80 el.
42.34
2.773
32.58
2.742
160 el.
2.876
3.88
2.64
3.625
For the sake of completeness, the proposed method’s performance was also compared to the conventional 3-node FE in the estimation of the first eigenmode, using 1 and 2 elements. In Figure 3, the first mode is presented for both models and mesh cases and is compared to the analytical mode. The presented results seem to be close to each other, so the absolute errors with respect to the analytical mode are illustrated in order to observe which model is more accurate.
Fig. 3
Prediction of rod’s mode 1 using the proposed trigonometric FE and the classic 3-node FE, employing a one element and b two elements. The red line represents the analytical mode 1
In Figure 4 the absolute errors of the first mode are presented for the proposed trigonometric FE and the conventional 3-node FE with respect to the analytical mode. It is obvious that the proposed method generally exhibits lower errors than the conventional 3-node FE, both with the utilization of one and two elements. However, it can be noticed that at the middle node location(s), the proposed elements yield higher errors than the classic 3-node FE. This happens due to the lack of continuity condition in this node’s shape function. Also, the errors in both models with two elements (Fig. 4b) are much lower than the one element discretization cases, for both models (Figure 4a). These results confirm the superiority of the interpolation capabilities of the proposed elements in free vibration analyses.
Fig. 4
Absolute errors of the prediction of the rod’s first mode by the proposed trigonometric FE and the classic 3-node FE with respect to the analytical mode solution, using a one element and b two elements
Finally, the convergence characteristics of the proposed elements are compared to the conventional quadratic FE in the estimation of the 20th eigenmode of the rod. The convergence approach described in [47, 48] is utilized, and the error between the predicted and analytical mode is calculated through the mean absolute error (MAE) metric. In Fig. 5 the MAE of each model is shown for each mesh case, and the convergence of both models towards the analytical solution is obvious. However, it can be observed that the conventional 3-node FE converge faster at the coarse discretization regime. This is not related with the interpolation properties of the proposed elements, since the interpolation superiority is already proven in the above case. The difference here is that the estimated mode is compared to the analytical mode only at the nodes’ locations, and it can be deduced that the middle (hidden) nodes of the proposed elements have lower estimation capabilities than the conventional 3-node FE, due to the lack of continuity condition of the middle node. Nonetheless, the existence of the middle nodes enhances the interpolation properties of the method, as proven in Fig. 4.
Fig. 5
Mean absolute error at the nodes of the proposed FE and conventional 3-node FE, in the estimation of the 20th mode of the rod
Similarly to the rod case, a clamped-pinned isotropic beam with geometric and material characteristics that are shown in Table 1 is considered for the modal analysis case study. The normalized ratios of numerical to analytical natural frequencies are plotted in Fig. 6 for the first 25 frequencies. The analytical solution for the natural frequencies is the one presented in the work of Chandrashekhara et al. [49]. Both the trigonometric and the conventional 3-node FE models utilize 28 elements. The analytical expression utilized to calculate the natural frequencies for a clamped-pinned beam from [49] is presented in Equation 15.
where \( a^2=\rho _AL^4NF^2/D_{11} \), \( b^2=\rho _D/\rho _AL^2 \) and \( c^2=D_{11}/A_{55}L^2 \) with \(NF\) being the beam’s natural frequencies. The natural frequencies are numerically calculated, and all the expressions mentioned above are presented in [49].
Fig. 6
Ratio of numerical to analytical natural frequencies (NF) for a clamped-pinned beam using conventional 3-node FE and the proposed trigonometric FE
It is obvious that the proposed trigonometric FE outperforms the classical 3-node in the majority of the domain of modes. Note that in the lower-frequency region, the conventional 3-node FE is closer to the analytical solution, but in the medium and high frequencies the trigonometric FE shows better performance. This fact also has an important impact on the allowed timestep for explicit integration, as also mentioned in subsection 3.1.1.
After this, a convergence investigation is accomplished for the estimation of the 20th natural frequency of the beam, following the procedure described in [47, 48]. The absolute error between the numerical and analytical 20th natural frequency and the respective convergence rates are presented in Table 3, for the conventional 3-node FE and the proposed trigonometric FE. It can be observed that the proposed trigonometric FE yield lower absolute errors in every mesh case, and the convergence rates of the two models are in a very close value range.
Table 3
Comparison of absolute errors and convergence rates of the conventional 3-node FE and proposed trigonometric FE, in the estimation of beam’s 20th natural frequency
Conventional 3-node FE
Proposed trigonometric FE
Mesh
Abs. Error
Conv. Rate
Abs. Error
Conv. Rate
20 el.
5294
–
5077
–
40 el.
605.2
3.129
520.7
3.285
80 el.
88.79
2.769
73.41
2.826
160 el.
6.067
3.871
5.577
3.718
As in the previous subsection, the prediction of the first normal mode in the same beam, but with clamped-free boundary conditions, is investigated. In Fig. 7 the simulated mode 1 of the two models is compared to the analytical mode 1. It is obvious that the two models yield similar results, but the proposed trigonometric FE results seem to be closer to the analytical mode 1.
Fig. 7
Prediction of beam’s mode 1 using the proposed trigonometric FE and the classic 3-node FE, employing (a) one element and (b) two elements. The red line represents the analytical mode 1
So, in order to validate this, in Fig. 8 the absolute errors of the proposed trigonometric FE and traditional 3-node FE with respect to the analytical first mode of the beam are depicted. Again, the proposed trigonometric FE exhibit lower errors than the classic 3-node FE in both discretization cases, verifying the selection of trigonometric shape functions for dynamic simulations.
Fig. 8
Absolute errors of the prediction of the beam’s first mode by the proposed trigonometric FE and the classic 3-node FE with respect to the analytical mode solution, using (a) one element and (b) two elements
At last, a convergence study of the proposed FE is performed via calculating the estimation errors in the estimation of the 20th normal mode of the beam. The convergence approach described in [47, 48] is employed, and the error between the predicted and analytical mode is calculated through calculation the MAE at the nodes. In Fig. 9 the MAE of each model is illustrated, where the convergence of both models towards the analytical solution is clear. However, it is shown that the conventional 3-node FE show lower errors at the coarse discretization regime. Again, this does not reduce the interpolation properties of the proposed FE inside the element space, but it is rather affected by the lower estimation capacity of the middle (hidden) node of the trigonometric FE that do not satisfy the continuity condition.
Fig. 9
Mean absolute error at the nodes of the proposed FE and conventional 3-node FE, in the estimation of the 20th mode of the beam
The explicit dynamic equilibrium for a conservative system with mass matrix \(\mathbf {[M]}\), stiffness matrix \(\mathbf {[K]}\), generalized displacements \(\mathrm {\underline{u}}\) and generalized force vector \(\mathrm {\underline{F}}\) is given as:
for timestep t. When using central differences, the acceleration \({\underline{\ddot{\textrm{u}}}}^t\) is expressed as \({\underline{\ddot{\textrm{u}}}}^t = (\mathrm {\underline{u}}^{t+\Delta t} - 2\mathrm {\underline{u}}^{t} + \mathrm {\underline{u}}^{t-\Delta t} ) \Delta t^{-2}\), where \(\Delta t\) is the timestep for the explicit integration scheme. Therefore, the generalized displacements are calculated at timestep \(t+ \Delta t\) as:
where \(a_0=\Delta t^{-2}\). Following the CFL condition, the timestep for each case in the next subsection is calculated using the relationship \(\Delta t=(4 \cdot NF_{max})^{-1}\), where \(NF_{max}\) is the maximum natural frequency.
3.2.1 Rod
In this first transient dynamic case study, a rod with the same geometric and material characteristics as in subsection 3.1.1 is considered. The rod is clamped-free and it is axially excited at its center (\(x=L/2\)) with a 5-cycle Hann-windowed pulse with 50 kHz central frequency. In Fig. 10, the snapshot of the rod’s wave response is illustrated at \(t=0.25 \ ms\). Two different models are compared, a 3-node conventional FE model and a model using the proposed trigonometric elements. It is obvious that the two models have an identical response with the utilization of 300 elements.
Fig. 10
Predicted axial displacement field of the rod (\(0.5 \ m \le x \le 3.5 \ m\)) at the end of the analysis
At this point, a convergence study of the aforementioned case study is performed for the direct comparison of the convergence rates between the proposed element and the traditional 3-node FE. The convergence study was pursued through consecutive simulations with increasing total element number, and the utilized error metric is given as:
where \(u_{n}^{x_i,t}\) is the displacement of spatial point \(x_i\) at timestep t for the model with a given mesh (n) and \(u_{n+1}^{x_i,t}\) is the displacement of spatial point \(x_i\) at the same timestep t for the model with denser mesh (\(n+1\)). In this study, the timestep t is the last timestep of each analysis; the initial mesh starts with 50 elements and goes up to 600 elements with a step of 50 elements. Of course, the displacement field is calculated at internal points (between nodes) in order to have the same vector size for every case.
The compared convergence results are presented in Fig. 11. It is obvious that the proposed trigonometric FE exhibit higher convergence rates than the traditional 3-node FE. In this way, fewer nodes are required when using the proposed FE in order to achieve an analysis with a given level of accuracy.
As already mentioned in subsection 3.1.1, the proposed trigonometric FE manifest lower values of the maximum natural frequency compared to the classic 3-node FE, so the value of the utilized timestep \(\Delta t\) for explicit integration solvers can be higher. In this way, the proposed elements can be significantly faster than the conventional 3-node FE for two reasons: i) the proposed trigonometric FE require less elements (or nodes) for the obtainment of an accurate solution (as in Fig. 11), and ii) the proposed trigonometric FE yield lower values of maximum natural frequencies (as in Fig. 2), thus permitting higher values of timesteps \(\Delta t\) in the explicit time integration procedure. So, in order to quantify the difference of those two models in terms of computing times, two simulations with same RMS error (\(RMS \ error \approx 1\)) with respect to Fig. 11 are performed. So, the proposed trigonometric FE model employs 550 elements, while the classic 3-node FE model utilized 600 elements. For completeness, it is mentioned that the proposed FE required a timestep \(\Delta t=5.58 \cdot 10^{-7} s\), while the classic 3-node FE required a timestep \(\Delta t=4.55 \cdot 10^{-7} s\). That means that the timestep of the proposed trigonometric FE model is about 23% higher than the one of the conventional 3-node FE model. Both simulations were conducted using MATLAB® R2019b with an Intel® Core i7-9750 H @ 2.60 GHz CPU and 16 GB RAM.
In Fig. 12, the CPU times for each model are presented. It is obvious that the proposed FE model is approximately 2.5 times faster than the traditional 3-node FE model. This crucial speed-up of the proposed model is a simultaneous result of its higher convergence rates and better prediction of the highest natural frequencies.
Fig. 11
Convergence plot of RMS error versus number of nodes for consecutive wave propagation simulations in a rod, using the proposed trigonometric FE and conventional 3-node FE
Continuing with the beam case, an aluminum beam with the geometric and material characteristics as in subsection 3.1.2 is taken into consideration. The beam is clamped-pinned, and it is transversely excited at its center (\(x=L/2\)) with a 5-cycle Hann-windowed pulse with 100 kHz central frequency. In Fig. 13, the last snapshot of the beam’s wave response is illustrated at \(t=0.15 \ ms\). Two different models are compared, a 3-node conventional FE model and a model using the proposed trigonometric elements. It is obvious that the two models have an identical response with the utilization of 500 elements.
Fig. 13
Predicted deflection of the aluminum beam (\(0.5 \ m \le x \le 1.5 \ m\)) at the end of the analysis
The computational effectiveness of the proposed trigonometric FE is presented in a convergence study of the analysis case described above. The simulation procedure consists of consecutive analyses with increasing mesh density. The utilized error metric is the one presented in Equation 19. As also depicted in the rod case, the proposed trigonometric FE manifest higher convergence rates than the conventional 3-node FE (Fig. 14).
Fig. 14
Convergence plot of RMS error versus number of nodes for consecutive wave propagation simulations in a beam, using the proposed trigonometric FE and conventional 3-node FE
In order to illustrate the higher computational efficiency of the proposed model, two analyses with equivalent accuracy (\(RMS \ error \approx 1\)) are conducted. The proposed FE model uses 900 elements, while the conventional 3-node FE model utilizes 1000 elements. The proposed FE required a timestep \(\Delta t=1.86 \cdot 10^{-7} s\), while the conventional 3-node FE required a timestep \(\Delta t=1.59 \cdot 10^{-7} s\). So, the timestep of the proposed model is about 17% higher than the timestep of the conventional 3-node FE. The explicit time durations of both simulations are displayed in Fig. 15. It can be observed that the proposed FE model is 1.7 times faster than the conventional 3-node FE model. It is expected that in more complex physical problems, this speed-up will be further increased due to the higher convergence rates of the proposed trigonometric FE.
Fig. 15
Bar chart of the explicit solver durations for two beam element models (classic 3-node FE and proposed trigonometric FE) of equivalent accuracy
The proposed elements, apart from their benefits, have also some limitations. A limitation that is manifested in this work is the lower estimation capabilities of the middle (hidden) nodes. It is shown that the middle nodes evince higher errors in the estimation of analytical eigenmodes compared to quadratic Lagrangian elements. However, the middle nodes increase the method’s interpolation capabilities inside the elemental space, as shown in Fig. 4 and 8. Another limitation of the proposed approach is that the hidden nodes should not be used to apply concentrated forces due to the value of the respective enrichment shape function shape on the hidden node (\(N_2(\xi =0)\ne 1\)). Apart from that, Gaussian quadratures cannot be employed, since the utilized shape functions are not polynomials. Additionally, the increased complexity of the trigonometric shape functions compared to simple polynomials, makes the consistent implementation of complex boundary conditions and loads more challenging.
An important aspect of a finite element is the shear locking effect [50, 51]. The shear locking is a common effect in linear shear deformable beam elements, where the structure appears to be stiffer than it should when the beam geometry (thickness, h, and length, L) is closer to the Euler-Bernoulli beam assumptions (\(L/h>20\)) rather than the Timoshenko beam assumptions (\(L/h<15\)) [52, 53]. In order to evaluate the existence of shear locking effect in the proposed elements, parametric modal and wave propagation simulations for several ratios L/h were performed, using both the proposed and the conventional 3-node FE, in the same case studies as in subsections 3.1.2 and 3.2.2. For the modal case 20 elements were used and for the wave propagation case 200 elements were used. Concerning the wave propagation process, in every simulation the maximum amplitude of the deflection is extracted at the end of the analysis, and then, these maximum amplitudes for both the proposed trigonometric FE and the classic 3-node FE are illustrated in Fig. 16. Since the maximum amplitudes of the two models coincide in the whole range of L/h ratios, and the quadratic FE do not exhibit shear locking, then it seems that the proposed FE do not suffer from shear locking in transient dynamic analyses, even for very high L/h ratios (\(L/h>>100\)) that are not usually met in practical applications.
Fig. 16
Maximum amplitudes of several wave propagation beam models using the proposed trigonometric and the conventional 3-node FE, for different ratios L/h
Concerning the modal analysis case study, the 1st and 20th natural frequency of the beam are estimated and compared for several ratios L/h, using the conventional 3-node FE and the proposed trigonometric FE (Fig. 17). It can be observed that the proposed element exhibits a level of shear locking, especially in the estimation of the 1st natural frequency, since the overestimation of the natural frequency means overestimation of the beam’s stiffness. It is also illustrated that the effect is not significant in the estimation of the 20th natural frequency, showing that shear locking appears in the lower-frequency range.
Fig. 17
Comparison of the estimated (a) 1st and (b) 20th natural frequency of the beam, using the proposed trigonometric FE and the conventional 3-node FE, for different ratios L/h
In this work, a new non-conventional trigonometric finite element with hidden nodes is proposed. The motivation for the selection of trigonometric functions is based on the analytical form of dynamic modeshapes that involve trigonometric functions. The selected trigonometric shape functions remind those of a traditional 3-node FE but do not expressly satisfy the continuity condition for shape functions. Specifically, the second shape function can be termed as an enrichment function since it is the one that does not explicitly satisfy the continuity condition but leads to the partition-of-unity property, and the second (middle) node can be termed as a hidden node. After the determination of the shape functions and their derivatives, two different element types are constructed: a rod element and a shear deformable Timoshenko beam element.
Since the rationale behind the selection of trigonometric functions is interconnected with dynamic structural response, modal and transient dynamic simulations of rods and beams are performed. It is highlighted that the systems that are studied are conservative systems without damping. The consideration of damping would not qualitatively change the results, so it is considered to exceed the scope of the current paper. Concerning the modal analyses, the proposed trigonometric rod element outperforms the conventional 3-node FE in terms of accuracy for every mode. When it comes to the beam element, the proposed FE exhibits better results than the traditional 3-node FE for all the studied frequencies except from the very low ones. In particular, both for rods and beams, the proposed trigonometric FE predict the highest natural frequencies with increased precision and lower overestimation compared to the classic 3-node FE. This fact is crucial in transient explicit dynamics since a higher timestep can be permitted according to the CFL condition. Regarding the transient dynamic simulations, both rod and beam trigonometric FE manifested higher convergence rates than the conventional 3-node FE in wave propagation analyses. Finally, with the synergy of the higher timestep requirement, the proposed trigonometric FE models lead to 1.7 and 2.5 faster simulations of wave propagation in beams and rods, respectively, compared to conventional 3-node FE. Future work may focus on the expansion of the proposed non-conventional FE for more complex structural problems, like two-dimensional dynamic simulations by developing 2D plane stress/strain elements or 2D plate elements.
Declarations
Conflict of interest
The authors declare no competing interests.
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All the integrals that are derived for the stiffness and mass matrices analytical solutions are numerically calculated. As mentioned in the main manuscript, the final stiffness and mass matrices involve a series of integrals between the terms of the shape functions and their derivatives. All the involved integrals are presented hereafter.
Substituting the constants \(\alpha , \ \beta , \ \gamma \) into the general form and changing the global variable x to the local \(\xi \), gives the final form:
Zienkiewicz, Olgierd Cecil., Taylor, Robert Leroy., Zhu, Jian Z.: The finite element method: its basis and fundamentals. Elsevier, 2005
2.
Timoshenko, S.: History of Strength of Materials: With a Brief Account of the History of Theory of Elasticity and Theory of Structures. Dover Civil and Mechanical Engineering Series. Dover Publications, 1983. isbn: 9780486611877
3.
Elishakoff, Isaac.: “Who developed the so-called Timoshenko beam theory?” Mathematics and Mechanics of Solids25(1) (2020), pp. 97–116
4.
Saravanos, D.A., Heyliger, P.R.: Coupled layerwise analysis of composite beams with embedded piezoelectric sensors and actuators. J. Intell. Mater. Syst. Struct. 6(3), 350–363 (1995)CrossRef
5.
Carrera, E.: Historical review of zig-zag theories for multilayered plates and shells. Appl. Mech. Rev. 56(3), 287–308 (2023)CrossRef
6.
Chrysochoidis, N.A., Saravanos, D.A.: Generalized layerwise mechanics for the static and modal response of delaminated composite beams with active piezoelectric sensors. Int. J. Solids Struct. 44, 8751–8768 (2007)CrossRef
7.
Runge, C., et al.: Über empirische funktionen und die interpolation zwischen äquidistanten ordinaten. Zeitschrift f ü r Mathematik und Physik 46(224–243), 20 (1901)
8.
Dauksher, W., Emery, A.F.: Accuracy in modeling the acoustic wave equation with Chebyshev spectral finite elements. Finite Elem. Anal. Des. 26(2), 115–128 (1997)CrossRef
9.
Hennings, B., Lammering, R., Gabbert, U.: Numerical simulation of wave propagation using spectral finite elements. CEAS Aeronaut. J. 4, 3–10 (2013)CrossRef
10.
Kudela, Paweł., Krawczuk, Marek., Ostachowicz, Wiesław.: “Wave propagation modelling in 1D structures using spectral finite elements”. In: Journal of sound and vibration 300.1-2 (2007), pp. 88–100
11.
Sprague, M.A., Geers, T.L.: Legendre spectral finite elements for structural dynamics analysis. Commun. Numer. Methods Eng. 24, 1953–1965 (2008)MathSciNetCrossRef
12.
Rekatsinas, C.S., et al.: A time-domain high-order spectral finite element for the simulation of symmetric and anti-symmetric guided waves in laminated composite strips. Wave Motion 53, 1–19 (2015)MathSciNetCrossRef
13.
Rekatsinas, C.S., Saravanos, D.A.: A hermite spline layerwise time domain spectral finite element for guided wave prediction in laminated composite and sandwich plates. J. Vib. Acoust. 139, 031009 (2017)CrossRef
14.
Eisenträger, Sascha., et al.: “On the numerical properties of high-order spectral (Euler-Bernoulli) beam elements”. In: ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f ü r Angewandte Mathematik und Mechanik 103.9 (2023), e202200422
15.
Hughes, Thomas JR., Cottrell, John A., Bazilevs, Yuri.: “Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement”. In: Computer methods in applied mechanics and engineering 194.39-41 (2005), pp. 4135–4195
16.
Nguyen, V.P., et al.: Isogeometric analysis: an overview and computer implementation aspects. Math. Comput. Simul. 117, 89–116 (2015)MathSciNetCrossRef
17.
Dvořáková, E., Patzák, B.: Isogeometric Bernoulli beam element with an exact representation of concentrated loadings. Comput. Methods Appl. Mech. Eng. 361, 112745 (2020)MathSciNetCrossRef
18.
Leung, A.Y., Tak., Au, Francis Tat Kwong,: Spline finite elements for beam and plate. Comput. Struct. 37(5), 717–729 (1990)
19.
Fan, SC., Cheung, YK.: “STATIC ANALYSIS OF RIGHT BOX GIRDER BRIDGES BY SPLINE FINITE STRIP MET-HOD.” In: Proceedings of the Institution of Civil Engineers 75.2 (1983), pp. 311–323
20.
Tham, LG., et al.: “Bending of skew plates by spline-finite-strip method”. In: Computers & structures 22.1 (1986), pp. 31–38
Dörfel, M.R., Jüttler, B., Simeon, B.: Adaptive isogeometric analysis by local h-refinement with T-splines. Comput. Methods Appl. Mech. Eng. 199, 264–275 (2010)MathSciNetCrossRef
23.
Uhm, Tae-Kyoung., Youn, Sung-Kie.: “T-spline finite element method for the analysis of shell structures”. In: International Journal for Numerical Methods in Engineering 80.4 (2009), pp. 507–536
24.
Guo, Mayi., et al.: “T-splines for isogeometric analysis of the large deformation of elastoplastic Kirchhoff–Love shells”. In: Applied Sciences 13.3 (2023), p. 1709
25.
Kuhn, L.M., et al.: Quintic and biquintic B-spline finite element solutions of clamped structures. Int. J. Comput. Methods Eng. Sci. Mech. 25, 463–474 (2024)
26.
Li, B., Chen, X.: Wavelet-based numerical analysis: a review and classification. Finite Elem. Anal. Des. 81, 14–31 (2014)MathSciNetCrossRef
27.
Majak, J., et al.: Higher-order Haar wavelet method for vibration analysis of nanobeams. Mater. Today Commun. 25, 101290 (2020)CrossRef
28.
Chen, X., et al.: Modeling of wave propagation in one-dimension structures using B-spline wavelet on interval finite element. Finite Elem. Anal. Des. 51, 1–9 (2012)MathSciNetCrossRef
Burgos, R.B., Santos, M.A., Cetale, et al.: Finite elements based on deslauriers-dubuc wavelets for wave propagation problems. Appl. Math. 7, 1490 (2016)
32.
Qiang, Yu., Hang, X., Liao, S.: Nonlinear analysis for extreme large bending deflection of a rectangular plate on non-uniform elastic foundations. Appl. Math. Model. 61, 316–340 (2018)MathSciNetCrossRef
33.
Wang, J., et al.: Multiresolution method for bending of plates with complex shapes. Appl. Math. Mech. 44, 561–582 (2023)MathSciNetCrossRef
Dimitriou, Dimitris K ., Nastos, Christos V., Saravanos, Dimitris A. “Multiresolution finite wavelet domain method for efficient modeling of guided waves in composite beams:”. In: Wave Motion 112 (2022), p. 102958
36.
Dimitriou, Dimitris K., Nastos, Christos V., Saravanos, Dimitris A.: “A multiresolution layerwise method with intrinsic damage detection capabilities for the simulation of guided waves in composite strips”. In: Journal of Vibration and Control (2023), p. 10775463231158667
37.
Olesch, D., et al.: Adaptive numerical integration of exponential finite elements for a phase field fracture model. Comput. Mech. 67, 811–821 (2021)MathSciNetCrossRef
38.
LaZghab, S., Aukrust, T., Holthe, K.: Adaptive exponential finite elements for the shear boundary layer in the bearing channel during extrusion. Comput. Methods Appl. Mech. Eng. 191, 1113–1128 (2002)CrossRef
39.
Schröppel, C., Wackerfuß, J.: Introducing the logarithmic finite element method: a geometrically exact planar Bernoulli beam element. Adv. Model. Simul. Eng. Sci. 3, 1–42 (2016)CrossRef
40.
Schröppel, C., Wackerfuß, J.: Co-rotational extension of the Logarithmic finite element method. PAMM 17, 345–346 (2017)CrossRef
41.
Heppler, G., Hansen, J.: “Performance of trigonometric basis function finite elements in Timoshenko beams”. In: 28th Structures, Structural Dynamics and Materials Conference. 1987, p. 843
42.
Heppler, G.R., Hansen, J.S.: Timoshenko beam finite elements using trigonometric basis functions. AIAA J. 26, 1378–1386 (1988)CrossRef
43.
Hansen, J.S., Heppler, G.R.: A mindlin shell element that satisfies rigid-body requirements. AIAA J. 23, 288–295 (1985)CrossRef
44.
Hashemi, SM., Richard, MJ., Dhatt, G.: “A new dynamic finite element (DFE) formulation for lateral free vibrations of Euler–Bernoulli spinning beams using trigonometric shape functions”. In: Journal of Sound and Vibration 220.4 (1999), pp. 601–624
45.
Strouboulis, Theofanis., Copps, Kevin., Babuška, Ivo.: “The generalized finite element method”. In: Computer methods in applied mechanics and engineering 190.32-33 (2001), pp. 4081–4193
46.
Lewy, H., Courant, R., Friedrichs, K.: Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100, 32–74 (1928)MathSciNetCrossRef
47.
Ratas, M., Majak, J., Salupere, A.: Solving nonlinear boundary value problems using the higher order Haar wavelet method. Mathematics 9, 21 (2021)CrossRef
48.
Arda, M., Majak, J., Mehrparvar, M.: Longitudinal wave propagation in axially graded Raylegh-Bishop nanorods. Mech. Compos. Mater. 59, 1109–1128 (2024)CrossRef
49.
Chandrashekhara, K., Krishnamurthy, K., Roy, Samit.: “Free vibration of composite beams including rotary inertia and shear deformation”. In: Composite Structures 14.4 (1990), pp. 269–279
50.
Prathap, G., Bhashyam, G.R.: Reduced integration and the shear-flexible beam element. Int. J. Numer. Meth. Eng. 18, 195–210 (1982)CrossRef
51.
Rakowski, J.: “The interpretation of the shear locking in beam elements”. In: Computers & Structures 37.5 (1990), pp. 769–776
52.
Averill, RC., Reddy, JN.: “Behaviour of plate elements based on the first-order shear deformation theory”. In: Engineering computations 7.1 (1990), pp. 57–74
53.
Wong, F., et al.: “Least-squares Smoothed Shape Functions for Constructing Field-Consistent Timoshenko Beam Elements”. In: Civil Engineering Dimension 27.1 (2025), pp. 22–32
