• Abstract

In this paper, by combining the modified one-leg $\theta$-methods with linear interpolation, a new class of one-leg $\theta$-methods for solving initial value problems of neutral pantograph equations with  multiple delay terms is presented. Our new approach, which is based on a geometric grid, exhibits better asymptotic stability compared to traditional $\theta$-methods. Under appropriate conditions, we prove, using the joint spectral radius method, that the proposed methods are asymptotically stable if and only if $0<\theta\le 1$. This indicates that the new methods overcome the limitation of the traditional $\theta$-methods, which require $\frac{1}{2}\le \theta\le1$ to ensure asymptotic stability. Furthermore, several numerical experiments are provided to validate our theoretical results.

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• Abstract

In this paper, we consider the dual continuum fluid filtration problem. The mathematical model is described by a coupled system of partial differential equations for two pressure fields. We propose a novel PINN-based partial learning approach to solve this problem. We employ a partially explicit time scheme, where the implicit scheme is applied to a highly permeable continuum, and the explicit scheme is used for the low permeable continuum. We utilize discrete-time physics-informed neural networks (PINNs) to compute the implicit part of the problem that is challenging to solve directly. By leveraging the slow variations in the low permeable continuum and the robustness of PINNs to data noise, we decouple the equations and solve them sequentially. At each time layer, we first solve the implicit part and then treat it as a known function to solve the explicit part. The explicit part is spatially discretized using the finite element method with standard linear basis functions. We utilize transfer learning to accelerate the computation of the implicit part using the parameters from the previous time layer as the initial parameters for training the current time layer. To test the proposed method, we consider two two-dimensional model problems. For each problem, the method achieved good accuracy, with transfer learning significantly speeding up the computation.

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This study investigates the dynamic response of functionally graded saturated porous (FGSP) nanoplates resting on the elastic foundation. The porosities are assumed to vary gradually through the thickness of the plate, following three different patterns: uniform, non-uniform symmetric, and non-uniform asymmetric distributions. Biot's poroelasticity theory is employed to describe the stress-strain relation of functionally graded porous materials when in a liquid-saturated state. Additionally, the nanoscale effects of the structure are taken into account by incorporating Eringen's nonlocal elasticity theory. The equations of motion are formulated by applying Hamilton's principle, utilizing a quasi-three-dimensional higher-order shear deformation (quasi-3D HSDT) theory. This theory guarantees that the top and bottom surfaces of the nanoplate experience conditions free of transverse shear stress. The obtained results reveal that the free vibration and transient response of the FGSP nanoplate is significantly influenced by various factors, including the porosity coefficient and distribution patterns, geometrical parameters, elastic foundation stiffness, Skempton coefficient, and nonlocal parameters. The theoretical development as well as numerical solutions presented herein offer valuable insights and serve as a reference for nonlocal theories of FGSP nanoplates.

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This paper proposes two efficient numerical schemes for the time dependent thermally coupled incompressible Magnetohydrodynamics (TMHD) equations. Firstly, the existence and uniqueness of the weak solutions are established by employing the Galerkin method and extracting the subsequences. Secondly, the Euler semi-implicit scheme is designed for the target problem. The energy dissipation and stability of the numerical scheme are developed, and the optimal error estimates are also presented. Thirdly, the implicit/explicit (IMEX) scheme is utilized to simplify the computational complexity, then the aimed problem splits into three linear elliptic subproblems with the constant coefficients at each time level $t_n$. Meanwhile, the "zero-energy contribution" (ZEC) approach is used to surmount the restriction $\Delta t\leq C$ caused by the IMEX scheme. The corresponding mathematical findings, including the energy dissipation and stability of the IMEX-ZEC scheme and the convergence of the numerical solutions are obtained via the energy method and the Gronwall lemma. Finally, we conduct some numerical results to confirm the established theoretical findings and show the performances of the considered numerical schemes.

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Plastic instability is an inherent property of metal materials, and enhancing the formability of sheet metal can be attributed to the retardation of plastic instability. In this study, we revisited the tensile instability of dual-phase (DP) steel by introducing a novel damage evolution model. Leveraging the theory of plastic instability and continuum damage mechanics (CDM), the instabilities of DP1180 steel sheet were investigated by examining the rotation angle of the uniaxial tensile specimen, which serves as an indicator for the onset of localized necking instability. Based on the hypothesis of elastic modulus equivalence, the damage evolution equation in uniaxial tension was derived mathematically, and the implicit and explicit expressions of the damage evolution were proposed. Furthermore, the relationship between effective stress and equivalent strain in the process of uniaxial tensile test was obtained. It was found that the sharp increase of effective stress was the direct factor leading to the fracture of DP1180 steel sheet. Finally, the mechanism of damage evolution was studied by scanning electron microscopy (SEM), including the nucleation, growth, and coalescence of voids. This study elucidates the mechanisms of tensile instabilities in DP steel sheets and provides a potential damage criterion for localized instability.

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• Abstract

In this paper, we propose two first-order, unconditionally stable, fully decoupled methods based on the Gauge-Uzawa method for solving the penalty Ericksen-Leslie system with variable density. The first method is based on the scalar auxiliary variable method for explicitly handling the nonlinear part, and the Gauge-Uzawa method for decoupling the velocity and pressure. The second method is formulated by introducing an auxiliary intermediate velocity variable for decoupling the director field computation from the velocity field, and then combining it with the Gauge-Uzawa method and the convex splitting method. Thus, two highly efficient, fully decoupled schemes are built. Furthermore, we use the finite element method to give an efficient implementation of the schemes. Then, we prove that both two schemes are unconditionally stable. Finally, numerical examples are provided to verify the convergence rate, unconditional stability, and computational efficiency. The proposed methods also provide a solution for establishing numerical schemes for other highly nonlinear variable-density coupled systems.

• General Summary