International Journal of Computational Methods

Finite mixture model is a useful probabilistic model for representing the probability distributions of observations. Gaussian mixture model (GMM) is a widely used one whose parameters are always estimated by the famous EM algorithm. But when probability distributions contain asymmetric modes, GMM requires much more constituent components to achieve a satisfied accuracy. Therefore, a generalized mixture finite model is proposed. First, it adopts the derivative $\lambda$-PDF which can well represent the asymmetric probability distributions as the probability density. However, the differential operation and solving the likelihood equations are scarcely possible in the M step of EM algorithm for the complex expression of the derivative $\lambda$-PDF. Second, a pseudo EM method is proposed to avoid the abovementioned difficulty which finds the pseudo maximum likelihood estimates of the parameters by utilizing the moment matching principle. It more easily estimates the parameters by solving a series of moment matching equations. Finally, four examples are presented to verify that the proposed generalized finite mixture model has advantage on representing the asymmetric probability distributions observations.

Layered materials play a crucial role in numerous essential structures, including adhesive connections, composite laminates, and various electronic components. This work focuses on proposing a novel framework of algorithms to improve the adaptability of element-free Galerkin method for modeling fracture in bimaterials exposed to thermoelastic loads. A bimaterial fracture problem is modeled as a combination of both weak and strong discontinuities — strong discontinuity owing to the presence of crack and weak discontinuity due to inherent material discontinuities. Initially, three distinct methods, comprising domain partitioning, lagrange multiplier, and jump function have been employed for interface crack modeling and error norm is used as an indicator to select the optimum one. Second, a novel interface enrichment algorithm has been introduced to effectively model the interface with limited computational costs with higher accuracy. Owing to the applicability of bimaterials in numerous applications under a combination of thermal/mechanical loads, the proposed formulation has been utilized for modeling and simulating a diverse bimaterial thermoelastic fracture problems using the proposed framework. The mixed mode (complex) stress intensity factors (SIFs) are numerically examined using modified domain form of interaction integral. A good agreement of results with available results from literature extends the computational prowess of the proposed framework in modeling a variety of thermoelastic bimaterial problems accurately and swiftly as compared to conventional EFGM.

Multi-material lattice structures demonstrate superior mechanical advantages and lightweight potential, enabling it preferable in many engineering fields, especially with the rise of additive manufacturing. However, employing multi-scale topology optimization inherently involves computational complexities of designing the macromaterial distribution and microlattice geometric size simultaneously. In this paper, a modified relative density mapping (MRDM) method is proposed, which generates a multi-material lattice structure by projecting the continuum multi-material topology optimization results. The proposed method extends the standard relative density mapping (RDM) method to multi-material problem and improves the performance. To achieve such purpose, a new mapping relation considering multi-material density and stress information is introduced to generate mapping weight of materials, which is the most novel contribution. Second, to avoid intersection of different materials, two types of material mapping strategies, namely continuous mapping strategy, and discrete mapping strategy, are introduced to generate multi-material lattice structure using the mapping weight of materials. Several numerical examples are exhibited, which demonstrate that the proposed method is capable of generating a multi-material lattice structure with clear material interface and good performance.

Particle Flow Code, as noncontinuous media calculation software, has found extensive applications in geotechnical engineering research. To simulate real physical processes, the commonly utilized servo-computing methods involve adjusting the boundaries to modify the model’s stress state. However, this approach alters the model’s initial boundaries, thereby compromising computational accuracy. Addressing this issue, this paper introduces a particle self-regulation servo-computing method. This method controls the internal particle self-adjustment and completes the servo through error analysis. On the basis of the error analysis of the average stress within the particle system and the servo stress during the iterative process, the method automatically calculates the change in particle size within the system, achieving rapid convergence to servo stress tolerance after finite number of iterations. Validation through uniaxial compression tests and complex slope cases indicates that this particle self-regulation servo method yields an error of less than 5% compared with conventional rigid servo methods. That is, it accurately achieves numerical simulation state control without altering the outer contours of the boundary model. The proposed computational method is applicable to servo calculations for irregular models in two or three dimensions, providing a viable approach for stress control in models under complex conditions.

Weather forecasting is an effort by meteorologists to predict whether states at a few prospective times and the conditions of weather that might be estimated. With new modern technology, present weather forecasting methods are highly precise. To achieve high accuracy, the methods developed for weather forecasting were much more complicated owing to many factors. Here, the usage of time series data for weather forecasting is done by the devised Long-Short Term Memory fused Convolutional neural network (LSTMFCNN). At first, the acquisition of input time series data from the specific dataset is done. In The feature extraction technical features are done by considering the input time series data. Then, the feature extraction is done utilizing the Rider Optimization Algorithm-Based Neural Network (RideNN) with the Soergel metric. RideNN is the integration of the Rider Optimization Algorithm (ROA) with the Neural Network (NN) classifier. Thus, the feature fusion step reduces the complexity and improves the accuracy. Thereafter, the oversampling technique is utilized for the data augmentation (DA) process. Finally, weather forecasting is done utilizing the newly designed LSTMFCNN and is obtained by the integration of Convolutional Neural Network (CNN) and Deep Long-Short Term Memory (DLSTM).

In this paper, methodological approaches for distribution functions of nonlinear stochastic differential equations (SDEs) with uncertain parameters are elaborated. The stationary and nonstationary Fokker–Planck–Kolmogorov (FPK) equations associated with the considered random equations are investigated. A semi-analytical approach based on the extended exponential closure handling the parameters uncertainty effects is presented. A new numerical method coupling the polynomial chaos with the differential quadrature method (DQM) is elaborated for the stationary case. A compact matrix formulation is established allowing considering a large number of generalized polynomial chaos. For nonstationary random FPK equations, the polynomial chaos is coupled with the quadrature discretization. A symmetric discretization of the main Fokker–Planck operator using a tensor product of the Hilbert state space and the Hilbert probabilistic space has resulted. The obtained eigenvalues and random eigenfunctions are used for a spectral decomposition. The time solution is obtained by the spectral expansion as well as by the random Euler stochastic method. The accuracy, effectiveness and advantages of the developed procedure in analyzing the stationary and nonstationary probabilistic solutions of nonlinear stochastic dynamical systems with uncertain parameters excited by a Gaussian white noise are demonstrated.

With the aid of the Caputo fractional derivative, this study constructs a novel mathematical model of magneto-thermoelasticity to investigate the transient phenomena for a fiber-reinforced rotating half-space under the influence of pulsed laser as heat source and inclined load in the context of refined three-phase-lag (TPL) Green–Naghdi (GN) models of generalized thermoelasticity, which is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. Together with the temperature gradient (Seebeck effect) and charge density influence, the generalized magneto-thermoelasticity model’s equations now include the modified Ohm’s law. An intense non-Gaussian laser beam heats the bounding plane’s surface. By using the normal mode analysis, the analytical expressions of the thermophysical quantities are produced in the physical domain. All physical quantities have been graphically depicted for different GN models (refined GN-II and GN-III models) to indicate the effect of memory-dependent derivative, Seebeck parameter, rotation and reinforcement of the medium under the influence of inclined load and pulsed laser.

The generalized Poisson–Boltzmann equation with the uniform ionic size is investigated. In mean-field theory, the ionic size effect is added by modifying the entropy effect of solvent molecules in the electrostatic free-energy functional of ion concentrations. Taking minimization of the mean-field electrostatic free-energy functional including the ionic size effect with respect to the ion concentrations, the minimum free-energy is obtained (see (4.6) in Li [(2009a) “Continuum electrostatics for ionic solutions with non-uniform ionic sizes,” Nonlinearity 22(4), 811–833]. Maximizing the minimum free-energy with respect to the electrostatic potential determines the generalized Poisson–Boltzmann equation. By the definition of shape derivative, the first variation of this minimum free-energy with respect to variation of the dielectric boundary is derived. Thus, the dielectric boundary force of an ionic solution with multiple ionic species with the uniform ionic size is obtained.

To develop a numerical method for global geophysical fluids, we usually need to choose a spherical grid and numerical approximations to represent the partial derivative equations. Some alternatives include the use of finite differences or finite volumes with latitude–longitude or reduced grids. Each of these cases has some advantages and also some limitations. This paper presents a comparison between two methods and describes a composite model using them side by side. The first is a well-known method for latitude–longitude grids and was used from 75${}^{\circ }$S until 75${}^{\circ }$N. The second is a recently developed scheme for reduced grids and was used only in the polar regions. The similarity between the two methods allows the use of small adaptations in their approximations to obtain consistency and mass conservation also in the transition between the two regions. The composite model combines advantages of the other two schemes and has a smaller computational cost. Numerical tests indicated order 2 of convergence, prevention of grid-imprinting errors, and avoidance of nonlinear instability. This model has numerical properties that may lead to efficient implementations with massive parallel computation.

This study proposes a method for calculating hyperelastic materials’ stress and tangent modulus using a generalized complex-step derivative approximation. This method is accurate and requires only the operational code of the strain energy density function. Moreover, because it uses only standard complex operations in Fortran, no prior preparation is required, making the method convenient and practical. The effectiveness of the method was confirmed by applying it to the implementation of isotropic and anisotropic hyperelastic models.

The corotational (CR) method is effective for extending linear elements to geometrically nonlinear analysis, but faces challenges with penalty elements based on modified couple stress theory due to C¹ continuity requirements. This paper presents an alternative CR formulation for geometrically nonlinear analysis of couple stress elasticity, approximating rigid body rotation incrementally. Then, a 4-node, 12-DOF plane element US-Q4$𝜃$-CS for size-dependent materials is extended for geometrically nonlinear analysis. Numerical examples demonstrate the new nonlinear element’s excellent performance in simulating size effects for the case of geometric nonlinearity.

The machining accuracy and cutting ability of abrasive suspension jet (ASJ) are affected by the spatial volume distribution and velocity changes of jet flow and abrasives in the free jet area outside the nozzle. In this paper, large eddy simulation (LES) flow model, volume of fluid (VOF) multiphase flow model and discrete element model (DEM) were used to simulate the flow field characteristics of ASJ. The spatial volume distribution and velocity changes of jet flow and abrasives under 10MPa jet pressure were studied and the model validation experiments were carried out. In the comparison between the numerical simulation results and the experimental results, it was found that the average error of the percentage of central jet flow is 3.4%, the percentage of central abrasive particles is 3.0% and the velocity of abrasive particles is 4.2%. The physical structure of the numerical model is fixed (so it does not support abrasive injection jet (AIJ)), and the material physical parameters, initial conditions and boundary conditions are editable. Therefore, it can be used to predict the degree of water divergence, abrasives divergence and abrasives velocity attenuation of ASJ with different jet pressure and abrasive types. This numerical model can provide guidance for the parameter setting of ASJ machining process in order to obtain better machining accuracy and cutting ability.

Kriging-based finite element method (K-FEM) is an enhancement of the FEM using Kriging interpolation in place of the conventional polynomial interpolation. This paper presents the development of the K-FEM for analysis of curved beams based on a deep arc theory, which accounts for membrane and shear deformation. The discrete strain gap technique is employed to circumvent shear locking and membrane locking. The numerical tests show that the Kriging-based elements are free from any locking, are able to provide highly accurate solutions with a regular course mesh discretization, and have excellent convergence characteristics, especially for circular beams.

In this paper, the newly-developed condensed Galerkin element of polynomial degree $\bar{m}$ for first-order initial-value problems is reformulated for and extended to the structural dynamic equations, yielding a high-performance dynamic element of one-step, unconditionally stable type with the conventional finite element (FE) convergence $O(h^{\bar{m}+1})$ within the element and the extra-superconvergence $O(h^{2\bar{m}+2})$ at the element end-nodes. Further, based on the element energy projection (EEP) technique, a superconvergence EEP formula for the condensed element is proposed in the paper and mathematical analysis is presented to prove that the derived EEP solution gains a superconvergence $O(h^{\bar{m}+2})$, which is an order higher than the FE solution of $O(h^{\bar{m}+1})$ for all element degree $\bar{m}\ge 1$ and is hence qualified to serve as a point-wise error estimator. As a result, a simple, efficient, and reliable algorithm with adaptive time-stepping controlled by the maximum norm is proposed. Representative numerical examples are presented to verify the validity of the proposed theorems and to demonstrate the high performance of the proposed element and the associated adaptivity algorithm.