International Journal of Computational Methods

Artificial neural networks (NNs) have been widely used in creation of various machine learning (ML) models. Training an NN for a given dataset is essentially finding a set of weights and biases, known as learning (or training) parameters. An NN uses the affine transformations to encode the features in the dataset in the form of learning parameters. One may naturally ask fundamental questions: (1) can the NN be trained, meaning that can a solution for a set of learning parameters can be found? and (2) is the solution unique? This paper addresses these questions. First, detailed formulation is presented for a simple NN with just one basic affine transformation unit (ATU) for linear regression problems. Conditions for solution existence and uniqueness will be examined, leading to the Solution Existence Theory. A discussion will be presented when this Solution Existence Theory is extended to general NNs for complex problems, in addition to a discussion on predictability of NNs.

The steady-state nonlinear problem governing the Newtonian flow of an incompressible viscous fluid in convection form in L-shaped domain is considered. The solution as $R_{\nu }$-generalized one in weighted sets is determined. Unlike the classical definition of the generalized solution in a weak formulation, which has a symmetric structure, our setting has an asymmetric one so that a special inf-sup condition in weighted sets is proved. A new weighted finite element scheme for an approximate $R_{\nu }$-generalized solution is constructed. A converging iterative method to solve the sequence of linear problems with block preconditioning of their matrices is introduced. The results of numerical simulations of several tests have shown an advantage over the standard methods so that an approximate $R_{\nu }$-generalized solution tends to the exact one with a rate almost twice in order exceed, relative to the grid step, than classical approaches. The result is achieved without using mesh refined in the neighborhood of a reentrant corner.

Multidisciplinary reliability analysis (MRA) is an important part of reliability-based multidisciplinary design optimization (RBMDO). In order to solve the problem of low computational efficiency caused by nested loops in multidisciplinary reliability analysis, an efficient multidisciplinary reliability analysis method based on decoupling idea is proposed. The method decouples system analysis, system sensitivity analysis and reliability analysis of Single discipline to form a recursive cycle, which eliminates the complete optimization calculation of reliability analysis model by system-level optimizer in each iteration. In the process of system sensitivity analysis, the global sensitivity equation is used to perform the concurrent subspace sensitivity analysis, which improves the efficiency of MRA. The decoupled multidisciplinary reliability analysis can get rid of the restriction that the traditional reliability analysis can only be carried out by using nonlinear optimization method, and the modified advanced mean value (MAMV) and other most probable point (MPP) search algorithms can be used to solve the reliability problem. Finally, three examples are used to verify the effectiveness of the method.

Cartilage damage and degeneration may lead to osteoarthritis for both animals and humans. Quantitative studies on the nonlinear hyper-elastic behavior of cartilages are essential to evaluate cartilage tissue deterioration. However, direct identification of the material behavior is not feasible. This paper presents a procedure to characterize the nonlinear mechanical behavior of the cartilage tissue by an inverse method using measurable structural quantities. First, a two-way neural network (NN) is established, which uses the fully trained forward problem neural network instead of the forward problem solver to generate training samples for inverse problem neural network. Moreover, based on the experimental data of the kangaroo shoulder joint, a nonlinear finite element (FE) model is then created to produce a dataset for training the forward network. Furthermore, intensive studies are conducted to examine the performance of our two-way NN method for the prediction of cartilage hyper-elastic material parameters by comparison with the direct inverse NN method. When only the direct inverse problem neural network is used for training, all samples are from FE simulations and the simulation time is 50.7 h, and the prediction time is tens of seconds. Besides, our two-way neural network calls the trained forward NN to collect training samples, and all the samples can be obtained in seconds, with which the simulation time is only 78 s. The predicted results are in good agreement with the experimental data, and the comparison shows that our two-way NN is an efficient and proficient method to predict the parameters for other biological soft tissues.

Due to the “overly-stiff” nature, it has been proved that the finite element method (FEM) is inevitable to suffer from the dispersion error when solving mid-frequency acoustic problems. In order to overcome this defect, in this study, a novel virtual node selection scheme of changing the virtual node’s position with a parameter is proposed, which is used to construct the condensed shape function of the cell-based radial point interpolation method (CS-RPIM) to soften the stiffness of discrete model. Based on the generalized gradient smoothing technique (GGST) and the condensed shape function technique, the present method provides a “close-to-exact” stiffness for discrete system model, which controls the dispersion error from the root. Several numerical examples have been researched to verify the performance of the proposed method in mid-frequency acoustic simulations. Moreover, with the excellent robustness, the present method lays a good foundation for solving complex practical engineering problems.

In this paper, we consider a system of large-scale nonlinear monotone equations and propose a class of derivative-free spectral conjugate gradient methods to solve it efficiently. We demonstrate the appropriate analytical features of this class and prove the global convergence theorems under a backtracking line search technique. In order to illustrate the numerical effectiveness of our class, we organize a competition in which 405 test problems will be solved by some members of the new class and four other similar derivative-free conjugate gradient methods. All the analytical and numerical results indicate that the presented class is promising.

This study investigates a new approach to solve a shear locking free on a low-order cell-centered finite element method (FECC) of the Reissner–Mindlin plate. The proposed method pursues the Lagrange piecewise of linear polynomial functions to attain the transverse displacement and rotations on the dual sub-mesh derived from the general one. The suggested theory is employed in numerical computation to verify the effectiveness and accurate performance. Detailed studies are carried out on several mesh types to compare with the analytical solutions reported by the low-order mimetic finite difference method [Beir ao Da Veiga, L., Lovadina, C. and Mora, D. [2013] “Numerical results for mimetic discretization of Reissner–Mindlin plate problems,” Calcolo 50, 209–237] and high-order mixed finite element methods [Chinosi, C. and Lovadina, C. [1995] “Numerical analysis of some mixed finite element methods for Reissner–Mindlin plates,” Comput. Mech. 16, 36–44]. The numerical results indicated that the present scheme based on the low-order FECC approximation can effectively predict on even the distorted meshes, satisfy the optimized convergence and be uniformly stable to the thin plate thicknesses.

Dielectric elastomers as a soft active material have been widely used in the field of artificial muscle actuator, acoustic actuator, loudspeaker, active control of vibration, soft robots and membrane resonators. Compared with traditional materials, there are many unknown uncertainties in the properties of the DE actuators. In this work, a viscoelastic dynamic model of dielectric elastomer is proposed with considering the uncertainties in material parameters, external mechanical load and voltage. By introducing the interval perturbation method and first-order Taylor series expansion method, the creep analysis, relaxation analysis and dynamic analysis of the dielectric elastomer with interval uncertain parameters are implemented. The effectivity of the proposed interval method is verified by the Monte Carlo simulation. This uncertain prediction method could be used in the design of active control systems with dielectric elastomers as actuators or sensors in the future.

The aim of this paper is to analyze the impacts of micromechanical approaches on the wave propagation of a functionally graded (FG) plate via indeterminate integral variables with hyperbolic secant shear displacement models. This model is established based on a high-order theory and a new displacement field with four unknowns introducing indeterminate integral variables with a secant hyperbolic shear function. Six micromechanical approaches are applied to approximate the effective material properties of an FG plate, namely Voigt’s model, Reuss’ model, Hashin–Shtrikman’s lower, and upper bound models, Tamura’s model, and the LRVE model. The volume fractions are supposed to change corresponding to the power-law and sigmoid. By applying Hamilton’s principle, general formulae of the wave propagation were obtained to get the wave modes and phase velocity curves of wave propagation in FG plates, with the impact of Voigt’s, Reuss’, Hashin–Shtrikman’s bounds, Tamura’s, and LRVE explicit micromechanical models.

In this paper, we derive new multi-parametric families of iterative methods whose orders range from six to eight, for solving nonlinear systems. Based on a generating function method known in the literature, we construct these families in the most general way possible in order to include some well-known methods as special cases. Several applied problems are solved to check the performance of our methods and other existing ones and to verify the theoretical results. It is found that our methods are competitive in performance compared to the other methods. Moreover, the basin of attraction method is introduced for nonlinear systems to confirm our findings and to choose the best performers.