International Journal of Computational Methods

In this paper, our focus is on the study of Hopf bifurcation in a diffusive predator–prey system that incorporates indirect predator-taxis. We commence by examining the stability of the unique positive constant steady state and the occurrence of Hopf bifurcation within this system. Following this, we utilize the center manifold theorem and the normal form theory to devise an algorithm for calculating the normal form of the Hopf bifurcation in this system. This algorithm enables us to determine the direction and stability of the resulting periodic solution from the Hopf bifurcation. Through numerical simulations, we demonstrate the effectiveness of our algorithm, thereby confirming the validity of our theoretical analysis. Additionally, we observe the emergence of stable spatially inhomogeneous periodic solutions resulting from the Hopf bifurcation.

A meshfree collocation method with particular reference to the lumped mass formulation is presented for the free vibration analysis of elastic solids. The lumped mass formulation leads to a simple unit entry for the collocation mass matrix. With the aid of the consistency conditions of meshfree approximation, a systematic theoretical investigation is provided to measure the frequency error of lumped mass meshfree collocation analysis of elastic solids, where both the pressure and shear waves are considered in the frequency accuracy order estimation. It is proven that the lumped mass meshfree collocation method yields similar convergence rates as those given by the consistent mass formulation, i.e., for a $p$th degree basis function, the frequency accuracy order is $p$ for even basis degree and $(p-1)$ for odd basis degree, respectively. In addition, it is shown that the lumped mass collocation approach using interpolatory meshfree shape functions is free of spurious modes. Numerical results consistently validate the proposed theoretical frequency error estimates for the free vibration analysis of elastic solids.

This paper proposes an improved nonprobabilistic Monte Carlo simulation method for structural reliability analysis based on convex models. The novelty of the proposed method is to use a unit hyper-sphere sampling technique to ensure the validity of all samples, and to restrict the sampling domain to the evaluated domain which is geometrically the intersection of a hyper-spherical shell and a hyper-spherical sector and generally has a smaller volume than the whole ellipsoid. The nonprobabilistic characteristics and application to the interval model of the proposed method are also discussed. Seven numerical examples are provided to demonstrate the efficacy of the proposed methodology.

This paper introduces a novel approach called the five-unknown trigonometric shear deformation (FTSD) theory for analyzing the flexural and buckling behavior of porous functionally graded (PFG) plates mounted on an elastic foundation. The FTSD theory features a unique trigonometric displacement field distribution that accurately captures the parabolic transverse shear stress distribution and satisfies the zero transverse shear stress conditions on the plate’s top and bottom surfaces while accounting for thickness stretching effects. The elastic foundation is modeled using a three-parameter Kerr-type model, consisting of two spring layers connected by a shear layer. The study considers both isotropic and sandwich PFG plates with various configurations of constituent layers, as well as both regular and irregular porosity types. The validity of the proposed methodology is evaluated by comparing its results with those from previously published research. Additionally, an extensive parametric analysis is conducted to explore the effects of porosity parameters and foundation parameters on the bending and buckling performance of PFG plates.

Addressing the demand for low-frequency noise control in ships and aiming to overcome the limitations imposed by the mass law of traditional materials, a sandwich thin-plate acoustic metamaterial (SPAM) was developed in this study. This structure is characterized by its simplicity and superior sound insulation performance. A sound insulation test was conducted to validate the exceptional low-frequency acoustic suppression capability of SPAM, and the effectiveness of the analytical model was also demonstrated. Subsequently, based on this model, a sensitivity analysis of the sound insulation characteristics of the SPAM relative to structural parameters was carried out. For the optimization process, the Logistic-Sine fusion chaotic mapping was employed to refine the dung beetle optimizer (DBO), enhancing the initial population distribution. The improved dung beetle optimizer (IDBO) was then used to further optimize and elevate the sound insulation performance of the SPAM. The outcomes indicate that the IDBO boasts superior optimization proficiency and accelerated convergence, while the optimized SPAM exhibits more remarkable sound insulation capabilities. Within the frequency range of 100–315 Hz, the average sound transmission loss (STL) has increased by 11.94 dB when compared to the pre-optimization values.

There has been very less work done on developing methods for solving systems of nonlinear equations using the idea of memorization. Methods without memory can be extended to methods with memory without using any additional function evaluations. Improvement in the convergence order of methods with memory gives an added benefit. The stability of such methods is another important concept which needs to be considered. In this work, we deal with the development of such methods. We first construct a sixth-order derivative free family without memory and then further extend it to methods with memory of order $7.40,7.87$ and $8.77$. Numerical results show the superiority of these methods over the existing ones. Basins of attraction have also been presented to observe the dynamical behavior of the proposed methods and the comparison validates that the proposed methods are much more stable than the existing methods.

This paper presents a hybrid approach to enhance heat transfer (HT) within a copper square duct for gas turbine (GT) blade cooling systems. The proposed method combines the gannet optimization algorithm (GOA) with the Spiking Deep Residual Network (SDRN), referred to as the GOA–SDRN technique. Since GTs operate at temperatures exceeding metal melting points, internal cooling is crucial to safeguard the blades and extend their lifespan. This study’s main goal is to apply rib tubular or turbulence promoters in internal cooling turbine blades, with a focus on managing temperature and maximizing HT to enhance GT blade-cooling systems. To achieve this, the proposed method analyzes the effectiveness of ribs by employing internal rib configurations, subsequently coated with composite nanomaterials such as thermal polymer and titanium carbide powder. The study investigates turbulent flow through the square passage at various Reynolds Numbers (Re), maintains constant ambient hot air temperature, and evaluates the performance of ribs. Specifically, the flow and HT within a square channel, resembling the air-cooled turbine blade cross-section, will be examined. The effectiveness of the GOA–SDRN method will then be assessed using MATLAB and compared against existing methodologies. The proposed method outperforms existing optimization methods significantly, achieving the highest efficiency of 94%, compared to 62% for SSA, 70% for CSA, and 84% for HBO. Additionally, the proposed method exhibits the lowest cost of $35, compared to $67 for SSA, $55 for CSA, and $48 for HBO. In summary, the proposed method offers superior efficiency and cost-effectiveness, making it the preferred choice for the application.

The principal objective of this paper is to propose a numerical scheme to solve linear and nonlinear third-kind Volterra integral equations (VIEs). The method approximates the solution using the collocation method based on moving least squares (MLS) approximation. The suggested scheme is meshless, since it needs no background mesh or cell structures, and is thus independent of the geometry of the domain. The approach reduces the solution of third-kind VIEs to the solution of systems of algebraic equations. All integrals appearing in the approach are calculated approximately by the Gauss–Legendre integration formula. The convergence analysis of the proposed scheme has been conducted, and its performance has been tested through a series of numerical examples, demonstrating its effectiveness and applicability in comparison to the existing methods.

Here, we present an investigation of the Cauchy problem solvability for the Laplace equation in a simply connected plane domain with a multi-component boundary. We reduce our investigation to solution of two singular integral equations. If the problem is resolvable, then we restore its solution via the integral Cauchy formula. We present the examples of the solvable problems. The construction involves an auxiliary approximate conformal mapping or the solution of certain singular integral equations.

The low-rank matrix approximation (LRMA) algorithm is an important method in signal processing. This algorithm is usually used in many tasks such as the matrix estimation and machine learning algorithms. In the past, many convex regularizers, such as the absolute-value norm, were presented for LRMA. Recently, many works show that LRMA on many nonconvex regularizers, such as the quadratic and logarithmic regularizers, can give better efficiency than the convex regularizer. Furthermore, many traditional works present LRMA in the univariate case, and do not consider in the multivariate case. Therefore, LRMA in the multivariate case on the nonconvex regularizer is presented in this work. Note that our proposed method is based on the singular value decomposition (SVD) algorithm. The relationship between singular values is considered by this multivariate case. We also present the novel nonconvex regularizer for LRMA. The simple solution for our method can be obtained from this regularizer. In many random signals, our proposed method is evaluated with the state-of-the-art algorithms. Experimental results show that the best results can be obtained from the proposed method.

We present here an uncertainty quantification methodology assisting fracture/damage detection that accounts for different sources of uncertainty involving in real-time damage identification. The relationship between structural responses (i.e., the natural frequencies) and the uncertain crack locations and lengths in mechanical systems is quantitatively assessed using variance-based sensitivity analysis. Sparse variational Gaussian process is employed to replace the computationally expensive finite element model. Dynamic behaviors of two mechanical systems which are (1) aluminum frame with adjustable masses and (2) cracked car suspension arm are analyzed through experimental and numerical data. The results demonstrate the effectiveness and robustness of combining these methods for structural assessments, providing precise and reliable insights applicable to complex mechanical models. We believe that this study enhances the understanding of vibration-induced fracture/damage and highlights the potential of this framework for advancing structural reliability analysis and dynamic system optimization.

The paper presents a robust collocation B-spline numerical scheme for solving singularly perturbed time lag parabolic differential–difference problems. The method uses Taylor’s series expansion for the approximation of retarded terms, the $𝜃$-method (when $𝜃=1$) for time discretization, and the extended third-degree B-spline collocation method for spatial discretization in a piecewise uniform Shishkin mesh. The advantage of using an extended cubic B-spline rather than the classical third-degree B-spline method is that it introduces one additional free parameter to control the global shape parameter. The stability and uniform convergence of the proposed scheme are investigated. The method is first-order accurate in $t$ and almost second-order accurate in $x$, surpassing existing literature methods.

Within the Discrete Element Method (DEM) framework, the multi-sphere model capable of creating irregular-shaped particles by several sub-spheres attracts increasing attention. Traditionally, the creation of irregular-shaped particles using the multi-sphere model requires manual determination of the position and diameter of each sub-sphere, which extremely increases the difficulty and workload of DEM modeling of irregular-shaped particles. To tackle this issue, a new automatic generation algorithm that helps to create arbitrary-shaped multi-sphere particles automatically was developed in this study. The quality of created multi-sphere particles can be easily controlled by adjusting the number of sub-spheres and size factor referred to as the size ratio of the smallest sub-sphere to the largest sub-sphere. To verify the effectiveness of the newly developed automatic generation algorithm, two verification tests were conducted in this study: (i) The packing behavior of irregular-shaped particles in a rectangular container and (ii) the mechanical behavior of irregular-shaped particles under the uniaxial compression. The fill heights of the simulations were in line with those of the experiments with an error of less than 8%. The mechanical behavior of irregular-shaped particles under the uniaxial compression can also be well reproduced. Besides, the influence of important parameters on simulation accuracy was also investigated. The results demonstrated that the newly developed automatic generation algorithm can accurately create complex-shaped particles with a small number of sub-spheres. Increasing the number of sub-spheres will improve the simulation accuracy but at the expense of computational efficiency. A larger size factor leads to higher computational efficiency in DEM simulation but is accompanied by a significant loss of accuracy.