• Abstract

In this paper, we present a Hessian recovery based linear finite element method to simulate the molecular beam epitaxy growth model with slope selection. For the time discretization, we apply a first-order convex splitting method and second-order Crank-Nicolson scheme. For the space discretization, we utilize the Hessian recovery operator to approximate second-order derivatives of a $C^0$ linear finite element function and hence the weak formulation of the fourth-order differential operator can be discretized in the linear finite element space. The energy-decay property of our proposed fully discrete schemes is rigorously proved. The robustness and the optimal-order convergence of the proposed algorithm are numerically verified. In a large spatial domain for a long period, we simulate coarsening dynamics, where 1/3- power-law is observed.

• General Summary

• Abstract

A Nitsche-based element-free Galerkin (EFG) method for solving semilinear elliptic problems is developed and analyzed in this paper. The existence and uniqueness of the weak solution for semilinear elliptic problems are proved based on a condition that the nonlinear term is an increasing Lipschitz continuous function of the unknown function. A simple iterative scheme is used to deal with the nonlinear integral term. We proved the existence, uniqueness and convergence of the weak solution sequence for continuous level of the simple iterative scheme. A commonly used assumption for approximate space, sometimes called inverse assumption, is proved. Optimal order error estimates in $L^2$ and $H^1$ norms are proved for the linear and semilinear elliptic problems. In the actual numerical calculation, the characteristic distance $h$ does not appear explicitly in the parameter $β$ introduced by the Nitsche method. The theoretical results are confirmed numerically.

• General Summary

• Abstract

A computational code is developed for the numerical solution of one-dimensional transient gas-liquid flows using drift-flux models, in isothermal and also with phase change situations. For these two-phase models, classical upwind schemes such as Roe- and Godunov-type schemes are generally difficult to derive and expensive to use, since there are no treatable analytic expressions for the Jacobian matrix, eigenvalues and eigenvectors of the system of equations. On the other hand, the high-order compact finite difference scheme becomes an attractive alternative on these occasions, as it does not make use of any wave propagation information from the system of equations. The present paper extends the localized artificial diffusivity method for high-order compact finite difference schemes to solve two-phase flows with discontinuities. The numerical method has simple formulation, straightforward implementation, low computational cost and, most importantly, high-accuracy. The numerical methodology proposed is validated by solving several numerical examples given in the literature. The simulations are sixth-order accurate and it is shown that the proposed numerical method provides accurate approximations of shock waves and contact discontinuities. This is an essential property for simulations of realistic mass transport problems relevant to operations in the petroleum industry.

• General Summary

• Abstract

This paper presents a specific network architecture for approximation of the first Piola-Kirchhoff stress. The neural network enables us to construct the constitutive relation based on both macroscopic observations and atomistic simulation data. In contrast to traditional deep learning models, this architecture is intrinsic symmetric, guarantees the frame-indifference and material-symmetry of stress. Specifically, we build the approximation network inspired by the Cauchy-Born rule and virial stress formula. Several numerical results and theory analyses are presented to illustrate the learnability and effectiveness of our network.

• General Summary

• Abstract

The meandering river is an unstable system with the characteristic of nonlinearity, which results from the instability of the flow and boundary. Focusing on the hydrodynamic nonlinearity of the bend, we use the weakly nonlinear theory and perturbation method to construct the nonlinear evolution equations of the disturbance amplitude and disturbance phase of two-dimensional flow in meandering bend. The influence of the curvature, $Re$ and the disturbance wave number on the evolution of disturbance amplitude and disturbance phase are analyzed. Then, the spatial and temporal evolution of the disturbance vorticity is expounded. The research results show: that the curvature makes the flow more stable; that in the evolution of the disturbance amplitude effected by curvature, $Re$ and the disturbance wave number, exist nonlinear attenuation with damping disturbances, and nonlinear explosive growth with positive disturbances; that the asymmetry distribution of the disturbance velocities increases with the curvature; that the location of the disturbance vorticity’s core area changes periodically with disturbance phase, and the disturbance vorticity gradually attenuates/increases with the decrease of the disturbance phase in the evolution process of damping/positive disturbances. These results shed light on the construction of the interaction model of hydrodynamic nonlinearity and geometric nonlinearity of bed.

• General Summary

• Abstract

In the hybrid RANS-LES simulations, proper turbulent fluctuations should be added at the RANS-to-LES interface to drive the numerical solution restoring to a physically resolved turbulence as rapidly as possible. Such turbulence generation methods mostly need to know the distribution of the characteristic length scale of the background RANS model, which is important for the recovery process. The approximation of the length scale for the Spalart-Allmaras (S-A) model is not a trivial issue since the model’s one-equation nature. As a direct analogy, the approximations could be obtained from the definition of the Prandtl’s mixing length. Moreover, this paper proposes a new algebraic expression to approximate the intrinsic length scale of the S-A model. The underlying transportation mechanism of S-A model are largely exploited in the derivation of this new expression. The new proposed expression is employed in the generation of synthetic turbulence to perform the hybrid RANS-LES simulation of canonical wall-bounded turbulent flows. The comparisons demonstrated the feasibility and improved performance of the new length scale on generating synthetic turbulence at the LES inlet.

• General Summary

• Abstract

In this paper, we present a linearized compact difference scheme for one-dimensional time-space fractional nonlinear diffusion-wave equations with initial boundary value conditions. The initial singularity of the solution is considered, which often generates a singular source and increases the difficulty of numerically solving the equation. The Crank-Nicolson technique, combined with the midpoint formula and the second-order convolution quadrature formula, is used for the time discretization. To increase the spatial accuracy, a fourth-order compact difference approximation, which is constructed by two compact difference operators, is adopted for spatial discretization. Then, the unconditional stability and convergence of the proposed scheme are strictly established with superlinear convergence accuracy in time and fourth-order accuracy in space. Finally, numerical experiments are given to support our theoretical results.

• General Summary

• Abstract

In this article, a new characteristic finite difference method is developed for solving miscible displacement problem in porous media. The new method combines the characteristic technique with mass-preserving interpolation, not only keeps mass balance but also is of second-order accuracy both in time and space. Numerical results are presented to confirm the convergence and the accuracy in time and space.

• General Summary

• Abstract

In this paper, we present a new vertex-centered arbitrary Lagrangian-Eulerian (ALE) finite volume scheme for two-dimensional compressible flow. In our scheme, the momentum equation is discretized on the vertex control volume, while the mass equation and the energy equation are discretized on the sub-cells which are included in the vertex control volume. We attain the average of the fluid velocity on the vertex control volume directly by solving the conservation equations. Then we can obtain the fluid velocity at vertex with the reconstructed polynomial of the velocity. This fluid velocity is chosen as the mesh velocity, which makes the mesh move in a Lagrangian manner. Two WENO (Weighted Essentially Non-Oscillatory) reconstructions for the density (the total energy) and the velocity are used to make our scheme achieve the anticipated accuracy. Compared with the general vertex-centered schemes, our scheme with the new approach for the space discretization can simulate some multi-material flows which do not involve large deformations. In addition, our scheme has good robustness, and some numerical examples are presented to demonstrate the anticipated accuracy and the good properties of our scheme.

• General Summary

• Abstract

In this paper, two fully-discrete local discontinuous Galerkin (LDG) methods are applied to the growth-mediated autochemotactic pattern formation model in self-propelling bacteria. The numerical methods are linear and decoupled, which greatly improve the computational efficiency. In order to resolve the time level mismatch of the discretization process, a special time marching method with high-order accuracy is constructed. Under the condition of slight time step constraints, the optimal error estimates of this method are given. Moreover, the theoretical results are verified by numerical experiments. Real simulations show the patterns of spots, rings, stripes as well as inverted spots because of the interplay of chemotactic drift and growth rate of the cells.

• General Summary

• Abstract

This study addresses the parameter identification problem in a system of time-dependent quasi-linear partial differential equations (PDEs). Using the integral equation method, we prove the uniqueness of the inverse problem in nonlinear PDEs. Moreover, using the method of successive approximations, we develop a novel iterative algorithm to estimate sorption isotherms. The stability results of the algorithm are proven under both a priori and a posteriori stopping rules. A numerical example is given to show the efficiency and robustness of the proposed new approach.

• General Summary

• Abstract

In this paper, a new cracked stiffener model for the stiffener with a part-through and open crack is proposed, considering the compatibility condition of displacements between the plate and the stiffener. Based on the first-order shear deformation theory, the free vibration of stiffened isotropic plates with cracked stiffeners are investigated for the first time. The description of the crack parameters is based on the continuous equivalent bending stiffness and equivalent depth of the cracked beam, and it takes into consideration of shear deformation, bending-extensional coupling vibration, and eccentricity between the stiffeners and the plate. The stiffened plates with single or multiple cracked stiffeners are formulated and discussed. The Ritz method with the modified characteristic functions is applied to demonstrate the effects of crack parameters (crack depth and location) coupling with the position and number of the cracked stiffeners on the vibration frequencies and modes of the stiffened plate. The validity and accuracy of the present solutions are verified through convergence studies and compared with the finite element results.

• General Summary

• Abstract

While the theory of peridynamics (PD) holds significant potential in engineering, its application is often limited by the significant computational costs by the nonlocality of PD. This research is based on a three-dimensional(3D) complex Timoshenko beam structure with six degrees of freedom. We propose a fast meshfree method based on the linear bond-based PD model of the stiffness matrix structure by ingeniously using the matrix decomposition strategy to maintain the Teoplitz structure of the stiffness matrix. This method significantly reduces the amount of calculation and storage without losing accuracy, reduces the amount of calculation from $\mathcal{O}(N^2)$ to $\mathcal{O}(N\log N)$, and decreases the storage capacity from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$. We validate the effectiveness of our approach through numerical examples, particularly in multi-beam structures. We demonstrate that our method realizes algorithm acceleration in numerical simulations of multi-beam structures subjected to static concentrated loads.

• General Summary

• Abstract

Helical waves are eigenfunctions of the curl operator and can be used to decompose an arbitrary three-dimensional vector field orthogonally. In turbulence study, high accuracy for helical waves especially of high wavenumber is required. Due to the difficulty in analytical formulation, the more feasible strategy to obtain helical waves is numerical computation. For circular cylinders of finite length, a semi-analytical method via infinite series to formulate the helical wave is known [E. C. Morse, J. Math. Phys., 46 (2005), 113511], where the eigenvalues are evaluated by iterating transcend equations. In this paper, the numerical computation for helical wave in a finite circular cylinder is implemented using a Chebyshev spectral method. The solving is transformed into a standard matrix eigenvalue problem. The large eigenvalues are computed with high precision, and the calculation cost to rule out spurious eigenvalues is significantly reduced with a new criterion suggested.

• General Summary

• Abstract

The present work contains an analytical expression and solution for free vibration problem of a composite lattice cylindrical shell surrounded by Winkler--Pasternak elastic foundation with clamped edges. The foundation is simulated using a large number of linear, homogenous shear and radial springs with variable stiffness. An integrated formula for calculation of the natural frequency of lattice structure and its foundation is derived from the equations of motion of the shell implemented by Winkler--Pasternak terms based on Fourier decomposition and Galerkin method. The fundamental frequency formula concerning the foundation elements and lattice parameters is an effective means of estimation frequency in earlier design phase and also a tool to assess the vibration analysis of composite lattice cylindrical shell surrounded by an elastic foundation in design analysis and numerical solutions. The results are verified and confirmed using finite element analysis which show a very good agreement.

• General Summary

• Abstract

In this paper two-dimensional differential quadrature method has been used to analyze thick Functionally Graded (FG) rotating disks with non-uniform boundary conditions and variable thickness. Material properties vary continuously along both radial and axial directions by a power law pattern.  Three-dimensional solid mechanics theory is employed to formulate the axisymmetric problem as a second order system of partial differential equations. The non-uniform boundary conditions are exerted directly into the governing equations  to reach the eigenvalue system of equations. Four different disk profile shapes are considered and discussed. The effect of the power law exponent is also investigated and results show that by the use of material which functionally varied along the radial and especially axial directions the stresses and strains can be controlled so the capability of the disk is increased. Comparison with other available approaches in the literature shows a good agreement here in terms of computational time, robustness and accuracy of the present method. Moreover, novel applications are shown  to provide results for further studies on the same topics.

• General Summary

• Abstract

Towards the solution reconstruction, one of the main steps in Godunov type finite volume scheme, a class of integrated linear reconstruction (ILR) methods has been developed recently, from which the advantages such as parameters free and maximum principle preserving can be observed. It is noted that only time-dependent problems are considered in the previous study on ILR, while the steady state problems play an important role in applications such as optimal design of vehicle shape. In this paper, focusing on the steady Euler equations, we will extend the study of ILR to the steady state problems. The numerical framework to solve the steady Euler equations consists of a Newton iteration for the linearization, and a geometric multigrid solver for the derived linear system. It is found that even for a shock free problem, the convergence of residual towards the machine precision can not be obtained by directly using the ILR. With the lack of the differentiability of reconstructed solution as a partial explanation, a simple Laplacian smoothing procedure is introduced in the method as a post-processing technique,  which dramatically improves the convergence to steady state. To prevent the numerical oscillations around the discontinuity, an efficient WENO reconstruction based on secondary reconstruction is employed. It is shown that the extra two operations for ILR are very efficient. Several numerical examples are presented to show the effectiveness of the proposed scheme for the steady state problems.

• General Summary

• Abstract

In this article, the effects of temperature and size-dependent on the buckling behavior of functionally graded (FG) cylindrical nanopanels resting on elastic foundation using nonlocal strain gradient theory are investigated in detail analytical approach. According to a simple power-law distribution, the material properties of FG cylindrical nanopanels are assumed to vary continuously through the thickness direction. The Pasternak model is used to describe the reaction of the elastic foundation on the FG cylindrical nanopanels. The fundamental relations and stability equations are derived by applying the nonlocal strain gradient theory and the classical shell theory based on the adjacent equilibrium criterion. Using Galerkin's method, the mechanical buckling behavior of FG cylindrical nanopanels resting on an elastic foundation in the thermal environment is solved. The reliability of the obtained results has been verified by comparison with the previous results in the literature. Based on the obtained results, the influences of the material length scale parameter, the nonlocal parameter, temperature increment, geometric parameters, material properties, and elastic foundation on buckling behaviors of FG cylindrical nanopanels resting on an elastic foundation in the thermal environment are analyzed and discussed.

• General Summary

• Abstract

This paper deals with nonlinear second-order initial value problems with time-variable delay. For solving this kind of problems, a class of implicit Runge-Kutta-Nystr{\"o}m (IRKN) methods with Lagrange interpolation are suggested. Under the suitable condition, it is proved that an IRKN method is globally stable and has the computational accuracy $\mathcal{O}(h^{\min\{p,\mu+\nu+1\}})$, where $p$ is the consistency order of the method and $\mu, \nu \ge  0$ are the interpolation parameters. Combining a fourth-order compact difference scheme with IRKN methods, an initial-boundary value problem of nonlinear delay wave equations is solved. The presented experiments further confirm the computational effectiveness of the methods and the theoretical results derived in previous.

• General Summary

• Abstract

In this study, a $(3+1)$ dimensional unstable gas flow system is applied and solved successfully via differential quadrature techniques based on various shape functions. The governing system of nonlinear four-dimensional unsteady Navier–Stokes equations of gas dynamics is reduced to the system of nonlinear ordinary differential equations using different quadrature techniques. Then, Runge-Kutta 4th order method is employed to solve the resulting system of equations. To obtain the solution of this equation, a MATLAB code is designed. The validity of these techniques is achieved by the comparison with the exact solution where the error reach to $\leq  1\times 10^{-5}$. Also, these solutions are discussed by seven various statistical analysis. Then, a parametric analysis is presented to discuss the effect of adiabatic index parameter on the velocity, pressure, and density profiles. From these computations, it is found that Discrete singular convolution based on Regularized Shannon kernels is a stable, efficient numerical technique and its strength has been appeared in this application. Also, this technique can be able to solve higher dimensional nonlinear problems in various regions of physical and numerical sciences.

• General Summary