• 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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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This article designs a new fifth-order finite volume mapped unequal-sized weighted essentially non-oscillatory scheme (MUS-WENO) for solving hyperbolic conservation laws on structured meshes. One advantage is that the new mapped WENO-type spatial reconstruction is a convex combination of a quartic polynomial with two linear polynomials defined on unequal-sized central or biased spatial stencils. Then we propose the new mapped nonlinear weights and new mapping function to decrease the difference between the linear weights and nonlinear weights. This method has the characteristics of small truncation errors and high-order accuracy. And it could give optimal fifth-order convergence with a very tiny $\varepsilon$ even near critical points in smooth regions while suppressing spurious oscillations near strong discontinuities. Compared with the classical finite volume WENO schemes and mapped WENO (MWENO) schemes, the linear weights can be any positive numbers on the condition that their summation is one, which greatly reduces the calculation cost. Finally, we propose a new modified positivity-preserving method for solving some low density, low pressure, or low energy problems. Extensive numerical examples including some unsteady-state problems, steady-state problems, and extreme problems are used to testify to the efficiency of this new finite volume MUS-WENO scheme.

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A shell whose height is far less than the minimum size covering the bottom is called the shallow shell. As a branch of linear elastic shell, it is a special shell with large span and has been widely applied in engineering fields. The main aim of this paper is to construct a general nonconforming finite element framework for a two-dimensional shallow shell model proposed by Ciarlet and Miara. Based on the different regularities of the displacement components, we give the special properties satisfied by the general framework and provide several nonconforming finite element discretization schemes. Then, the existence and uniqueness of the numerical solutions are proved, with the rate of convergence derived. Finally, numerical experiments are carried out for the paraboloid, spherical dome and cylindrical bridge, which validates the theoretical analyses. Moreover, the computing cost of discretizing the shallow shell model is evidently less than that of discretizing the general shell model with comparable accuracy when the shell is the large span shell.

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In the present study, a multi-pseudo-potential model is used to simulate the deformation and breakup of bubbles and droplets under simple shear flow. It is shown that the current model can adjust the amount of surface tension, independent of the interface thickness, equation of state (EOS), and reduced temperature. Considering the available findings, no comprehensive study has been performed on all aspects of deformation of bubbles and droplets under shear flow using numerical methods. Bubble or droplet deformation under simple shear flow depends on two non-dimensional numbers: capillary number (Ca) and viscosity ratio $(\lambda)$. In this investigation, various scenarios, including small deformation, large deformation, and breakup for bubbles $(0.2 < \lambda < 1)$ and droplets $(1 < \lambda < 5)$, are separately studied. Results of the multi-pseudo-potential model show that the bubble and droplet deformations oscillate under shear flow and undergo elongation and contraction over time to converge to the final shape. As the capillary increases by more than one, the bubble expands and shrinks. The bubble tips become sharp, and so-called slender, at a low viscosity ratio $(\lambda)$. A detailed comparison is made between the numerical results for deformation parameters of the present model and the experimental results available in the references.

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