Search Results (87)

Nowadays, guided acoustic waves (GAW) are used for many sensor and actuator applications. The use of numerical methods can facilitate the development and optimization process enormously. In this work, a universally applicable finite element method (FEM)-based model is introduced to determine the dispersion relations of guided acoustic waves. A 2-dimensional unit cell model with Floquet periodicity is used to calculate the corresponding band structure diagrams. Starting from a free plate the model is expanded to encompass single-sided fluid loading. Followed by a straightforward algorithm for post-processing, the data is presented. Additionally, a parametric optimizer is used to adapt the simulations to experimental data measured by a laser Doppler vibrometer on an aluminum plate. Finally, the accuracy of the FEM model is compared to two reference models, achieving good consistency. In the case of the fluid-loaded model, the behavior of critical interactions between the dispersion curves and model-based artifacts is discussed. This approach can be used to model 2D structures like phononic crystals, which cannot be simulated by common GAW models. Moreover, this method can be used as input for advanced multiphysics simulations, including acoustic streaming applications. Full article

This study presents a hyperbolic three-dimensional telegraph interface model with regular interfaces, numerically solved using a hybrid scheme that integrates Haar wavelets and the finite difference method. Spatial derivatives are approximated via a truncated Haar wavelet series, while temporal derivatives are discretized using the finite difference method. For linear problems, the resulting algebraic system is solved using Gauss elimination; for nonlinear problems, Newton’s quasi-linearization technique is applied. The method’s accuracy and stability are evaluated through key performance metrics, including the maximum absolute error, root mean square error, and the computational convergence rate $R_c\left(M\right)$, across various collocation point configurations. The numerical results confirm the proposed method’s efficiency, robustness, and capability to resolve sharp gradients and discontinuities with high precision. Full article

Material defects resulting from manufacturing and processing can significantly affect material properties. Voids and dislocations are material defects considered in this study, in which a numerical solution of the 3D stress field of a dislocation line (infinite or finite) outside a cylindrical void (either parallel to the cylinder axis or not) is developed using the collocation point method. The collocation point method is utilized to solve ordinary differential equations, partial differential equations, differential-algebraic equations, and integral equations by enforcing the solution at a set of spatial collocation points. Analytical solutions for such three-dimensional (3D) problems, e.g., a dislocation line or segment near an internal void of any shape, were not found. Therefore, a numerical solution for this problem has been constructed in this paper. The numerical solution developed is verified using an existing two-dimensional analytical solution. The numerical results and the 2D analytical solution are in perfect agreement as long as the cylindrical void is sufficiently long and the Saint-Venant’s principle is followed. Full article

Regarding the discretization issue in the process of a simultaneous approach for dynamic optimization problems, a configuration strategy based on the density function has been proposed for the finite element distribution of dynamic optimization problems. By utilizing the error at the non-collocation points, a bilevel problem has been constructed and solved, and the number and distribution of the finite elements have been evaluated. For the inner problem in the bilevel problem, a new smoothing function has been introduced to improve the solution accuracy of the inner problem. The grid density function has been constructed using the error at the non-collocation points on each finite element. Finally, the grid density function has been used to update the positions of the finite element endpoints, with the aim of reallocating the finite elements. Finally, two case studies have been provided to specifically demonstrate the role of the proposed method in reducing the optimization problem scale with required solution accuracy. Full article

This paper uses the flexural vibration of cantilever beams as a benchmark problem to test mesh-free and finite element methods in structural dynamics. First, a symbolic analysis of the “kernel collocation” type mesh-free method is carried out, in which the collocation function satisfies the boundary conditions. This enables both Finite Element (FE) and mesh-free results to be compared with exact analytical ones. Thereafter, the natural frequencies and Frequency Response Function (FRF), in terms of the beam parameters, are determined and compared with the analytical results, that exist in the literature. It is shown that by adjusting the parameters of the kernel function, we can find identical peaks to those of the analytical method. The finite element method is also employed to solve this problem, and the first three natural frequencies were computed in terms of the beam parameters. When comparing the two methods, we see that by increasing the number of elements in the FEM we can always achieve better accuracy, but we will obtain twice the number of modal frequencies. However, the mesh-free method with the same number of nodes does not provide these extra frequencies. From this benchmark problem, it is concluded that the accuracy of the mesh-free methods always depends on the adjustment of the kernel function. However, the FEM is advantageous because it does not require such adjustments. Full article

In many tasks of railway vibration, the structure, that is, the track, a bridge, and a nearby building and its floors, is coupled to the soil, and the soil–structure interaction and the damping by the soil should be included in the analysis to obtain realistic resonance frequencies and amplitudes. The stiffness and damping of a variety of foundations is calculated by an indirect boundary element method which uses fundamental solutions, is meshless, uses collocation points on the boundary, and solves the singularity by an appropriate averaging over a part of the surface. The boundary element method is coupled with the finite element method in the case of flexible foundations such as beams, plates, piles, and railway tracks. The results, the frequency-dependent stiffness and damping of single and groups of rigid foundations on homogeneous and layered soil and the amplitude and phase of the dynamic compliance of flexible foundations, show that the simple constant stiffness and damping values of a rigid footing on homogeneous soil are often misleading and do not represent well the reality. The damping may be higher in some special cases, but, in most cases, the damping is lower than expected from the simple theory. Some applications and measurements demonstrate the importance of the correct damping by the soil. Full article

In this study, a class of nonlinear heterogeneous reaction–diffusion system (RDS) has been considered that arises in modeling epidemiological interactions, environmental sciences, and chemical and ecological systems. Numerical and analytic solutions for this kind of variable medium nonlinear RDS are challenging. This article developed a few highly accurate numerical schemes for such problems. For the spatial integration of the heterogeneous RDS, a few finite difference schemes, a Bernstein collocation scheme, and a Fourier transform scheme were employed. The stability and accuracy analysis of the spectral schemes were studied to confirm the order of convergence of the approximation. A few methods of lines were then used for the temporal integration of the resulting semidiscrete model. It was confirmed theoretically that the spectral/pseudo-spectral method is very efficient and highly accurate for such a model. A fast and efficient solver for the resulting full discrete system is highly desired. A Newton–GMRES–Multigrid solver was applied for the resulting full discrete system. It is demonstrated in tabular form that a multigrid accelerated Newton–GMRES solver outperforms most linear solvers for such a model. Full article

This work aims to provide approximate solutions for singularly perturbed problems with periodic boundary conditions using quintic B-splines and collocation. The well-known Shishkin mesh strategy is applied for mesh construction. Convergence analysis demonstrates that the method achieves parameter-uniform convergence with fourth-order accuracy in the maximum norm. Numerical examples are presented to validate the theoretical estimates. Additionally, the standard hybrid finite difference scheme, a cubic spline scheme, and the proposed method are compared to demonstrate the effectiveness of the present approach. Full article

This article investigates an axisymmetric contact problem involving the interaction between a rigid cylindrical stamp and a poroelastic cylinder of finite dimensions, based on the Cowin–Nunziato theory of media with voids. The stamp is assumed to have a flat base and to be in frictionless contact with the cylinder. The cylinder, in turn, rests on a rigid base without friction, with no normal displacements or tangential stresses on its lateral surface. Under an applied vertical force, the stamp undergoes displacement, compressing the poroelastic cylinder. The mathematical formulation of this problem involves expressing the unknown displacements within the cylinder and the variation in pore volume fraction as a series of Bessel functions. This representation reduces the problem to an integral equation of the first kind, describing the distribution of contact stresses beneath the stamp. The kernel of the integral equation is explicitly provided in its transformed form. The collocation method is employed to solve the integral equation, enabling the determination of contact stresses and the relationship between the indenter’s displacement and the applied force. A comparative model parameter analysis is performed to examine the effects of different material porosity parameters and model geometrical characteristics on the results. Full article

A fully discrete computational technique involving the implicit finite difference technique and cubic Hermite splines is proposed to solve the non-linear conformable damped Burgers’ equation with variable coefficients numerically. The proposed scheme is capable of solving the equation having singularity at $t=0$. The space direction is discretized using cubic Hermite splines, whereas the time direction is discretized using an implicit finite difference scheme. The convergence, stability and error estimates of the proposed scheme are discussed in detail to prove the efficiency of the technique. The convergence of the proposed scheme is found to be of order $h^2$ in space and order ${(\Delta t)}^{\alpha }$ in the time direction. The efficiency of the proposed scheme is verified by calculating error norms in the Eucledian and supremum sense. The proposed technique is applied on conformable damped Burgers’ equation with different initial and boundary conditions and the results are presented as tables and graphs. Comparison with results already in the literature also validates the application of the proposed technique. Full article

In this paper, we have established a numerical method for a class of time-fractional and Riesz space distributed-order advection–diffusion equation with time-delay. Firstly, we transform the Riesz space distributed-order derivative term of the diffusion equation into multi-term fractional derivatives by using the Gauss quadrature formula. Secondly, we discretize time by using second-order finite differences, discretize space by using second kind Chebyshev polynomials, and convert the multi-term fractional equation to a system of algebraic equations. Finally, we solve the algebraic equations by the iterative method, and prove the stability and convergence. Moreover, relevant examples are shown to verify the validity of our algorithm. Full article

The failure of overhead transmission lines in the United States can lead to significant economic losses and widespread blackouts, affecting the lives of millions. This study focuses on analyzing the failure of transmission lines, specifically considering the effects of wind, ambient temperature, and current demands, incorporating minimal and significant pre-existing damage. We propose a multiphysics framework to analyze the transmission line failures across sensitive and affected states of the United States, integrating historical data on wind and ambient temperature. By combining numerical simulation with historical data analysis, our research assesses the impact of varying environmental conditions on the reliability of transmission lines. Our methodology begins with a deterministic approach to model temperature and damage evolution, using phase-field modeling for fatigue and damage coupled with electrical and thermal models. Later, we adopt the probability collocation method to investigate the stochastic behavior of the system, enhancing our understanding of uncertainties in model parameters, conducting sensitivity analysis to identify the most significant model parameters, and estimating the probability of failures over time. This approach allows for a comprehensive analysis of factors affecting transmission line reliability, contributing valuable insights into improving power line’s resilience against environmental conditions. Full article

Anomalous diffusion of particles has been described by the time-fractional reaction–diffusion equation. A hybrid formulation of numerical technique is proposed to solve the time-fractional-order reaction–diffusion (FRD) equation numerically. The technique comprises the semi-discretization of the time variable using an L1 finite-difference scheme and space discretization using the quintic Hermite spline collocation method. The hybrid technique reduces the problem to an iterative scheme of an algebraic system of equations. The stability analysis of the proposed numerical scheme and the optimal error bounds for the approximate solution are also studied. A comparative study of the obtained results and an error analysis of approximation show the efficiency, accuracy, and effectiveness of the technique. Full article

Motion control is one of the three core modules of autonomous driving, and nonlinear model predictive control (NMPC) has recently attracted widespread attention in the field of motion control. Vehicle dynamics equations, as a widely used model, have a significant impact on the solution efficiency of NMPC due to their stiffness. This paper first theoretically analyzes the limitations on the discretized time step caused by the stiffness of the vehicle dynamics model equations when using existing common numerical methods to solve NMPC, thereby revealing the reasons for the low computational efficiency of NMPC. Then, an A-stable controller based on the finite element orthogonal collocation method is proposed, which greatly expands the stable domain range of the numerical solution process of NMPC, thus achieving the purpose of relaxing the discretized time step restrictions and improving the real-time performance of NMPC. Finally, through CarSim 8.0/Simulink 2021a co-simulation, it is verified that the vehicle dynamics model equations are with great stiffness when the vehicle speed is low, and the proposed controller can enhance the real-time performance of NMPC. As the vehicle speed increases, the stiffness of the vehicle dynamics model equation decreases. In addition to the superior capability in addressing the integration stability issues arising from the stiffness nature of the vehicle dynamics equations, the proposed NMPC controller also demonstrates higher accuracy across a broad range of vehicle speeds. Full article

The thermal analysis of brake discs is crucial for studying issues such as wear and cracking. This paper establishes a symmetric two-dimensional brake disc model using the barycentric rational interpolation collocation method (BRICM). The model accounts for the effects of thermal radiation and is linearized using Newton’s linear iteration method. In the spatial dimension, the two-dimensional heat conduction equation is discretized using BRICM, while in the temporal dimension, it is discretized using the finite difference method (FDM). The resulting temperature distribution of the brake disc during two consecutive braking events is consistent with experimental data. Additionally, factors affecting the accurate calculation of the temperature are examined. Compared to other models, the proposed model achieves accurate temperature distributions with fewer nodes. Furthermore, the numerical results highlight the significance of thermal radiation within the model. The results obtained using BRICM can be used to predict the two-dimensional temperature distribution of train brake discs. Full article