Fast solvers of integral and pseudodifferential equations on closed curves

On the basis of a fully discrete trigonometric Galerkin method and two grid iterations we propose solvers for integral and pseudodifferential equations on closed curves which solve the problem with an optimal convergence order , (Sobolev norms of periodic functions) in arithmetical operations.

     

A Spatial Geometric Nonlinearity Spline Beam Element With Nodal Parameters Containing Strains北大核心CSCD

Many slender rods in engineering can be modeled as Euler-Bernoulli beams. For the analysis of their dynamic behaviors, it is necessary to establish the dynamic models for the flexible multi-body systems. Geometric nonlinear elements with absolute nodal coordinates help solve a large number of dynamic problems of flexible beams, but they still face such problems as shear locking, nodal stress discontinuity and low computation efficiency. Based on the theory of large deformation beams’ virtual power equations, the functional formulas between displacements and rotation angles at the nodes were established, which can satisfy the deformation coupling relationships. The generalized strains to describe geometric nonlinear effects in this case were derived. Some parameters of boundary nodes were replaced by axial strains and sectional curvatures to obtain a more accurate and concise constraint method for applying external forces. To improve the numerical efficiency and stability of the system’s motion equations, a model-smoothing method was used to filter high frequencies out of the model. The numerical examples verify the rationality and effectiveness of the proposed element. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

On multivariate polynomial interpolation

We provide a map which associates each finite set in complexs-space with a polynomial space from which interpolation to arbitrary data given at the points in is possible and uniquely so. Among all polynomial spacesQ from which interpolation at is uniquely possible, our is of smallest degree. It is alsoD- and scale-invariant. Our map is monotone, thus providing a Newton form for the resulting interpolant. Our map is also continuous within reason, allowing us to interpret certain cases of coalescence as Hermite interpolation. In fact, our map can be extended to the case where, with eachgq, there is associated a polynomial space P, and, for given smoothf, a polynomialqQ is sought for which

In what sense is the rational interpolation problem well posed

It is well known that the nonlinear problem of interpolatingm+n+1 data by a rational function of type (m, n) may have no solution, but that the corresponding linearized problem (obtained by multiplying through by the denominator) always leads to a unique rational function, which is often still called the rational interpolant. For fixedm andn, and fixed (possibly multiple) interpolation points, the dependence of this interpolant on the prescribed function values is studied here. For ten notions of convergence in the space _{m, n} the question of the continuity of this interpolation operator is investigated.Communicated by William B. Gragg.AMS classification: 41A24, 30E05, 41A20, 65D05.     

Mass conservation in computational morphodynamics: uniform sediment and infinite availability

Computational morphodynamics in finite volume methods are based on the evaluation of the rate of bed level change in the vertices on the deforming bed. With the use of finite volume methods on collocated (unstructured) grids, the rate of bed level change needs to be interpolated from the mesh faces to the vertices. First, this work reviews two methods based on a vectorial shape of the bed evolution equation (no scalar contributions from storage, erosion and deposition) in terms of their mass conserving properties. Second, a method that allows for scalar contributions in the bed evolution equation (the Exner equation) is proposed for general, unstructured meshes, and an analytical derivation for the simple one‐dimensional problem on a non‐equidistantly discretised grid is considered. The solution is compared with the general two‐dimensional formulation. The two‐dimensional formulation leads to the formulation of a geometric sand sliding routine on unstructured grids. The newly proposed interpolation method and the sand sliding routine are tested, and mass conservation of the sediment is considered with special emphasis on the effect of the solution accuracy for the suspended sediment transport. Discussions on other interpolation methods and their mass conserving properties are given with a special focus of the distance weighted interpolation method directly available and easily applied in O penFOAM . Furthermore, effects from horizontal displacements of the vertices, explicit filtering of the evolving bed and morphological acceleration on global mass conservation, are discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

A mixed‐interpolation finite element method for incompressible thermal flows of electrically conducting fluids

A new mixed‐interpolation finite element method is presented for the two‐dimensional numerical simulation of incompressible magnetohydrodynamic (MHD) flows which involve convective heat transfer. The proposed method applies the nodal shape functions, which are locally defined in nine‐node elements, for the discretization of the Navier–Stokes and energy equations, and the vector shape functions, which are locally defined in four‐node elements, for the discretization of the electromagnetic field equations. The use of the vector shape functions allows the solenoidal condition on the magnetic field to be automatically satisfied in each four‐node element. In addition, efficient approximation procedures for the calculation of the integrals in the discretized equations are adopted to achieve high‐speed computation. With the use of the proposed numerical scheme, MHD channel flow and MHD natural convection under a constant applied magnetic field are simulated at different Hartmann numbers. The accuracy and robustness of the method are verified through these numerical tests in which both undistorted and distorted meshes are employed for comparison of numerical solutions. Furthermore, it is shown that the calculation speed for the proposed scheme is much higher compared with that for a conventional numerical integration scheme under the condition of almost the same memory consumption. Copyright © 2016 John Wiley & Sons, Ltd.     

Acoustic simulation using a novel approach for reducing dispersion error

It is well‐known that the traditional finite element method (FEM) fails to provide accurate results to the Helmholtz equation with the increase of wave number because of the ‘pollution error’ caused by numerical dispersion. In order to overcome this deficiency, a gradient‐weighted finite element method (GW‐FEM) that combines Shepard interpolation and linear shape functions is proposed in this work. Three‐node triangular and four‐node tetrahedral elements that can be generated automatically are first used to discretize the problem domain in 2D and 3D spaces, respectively. For each independent element, a compacted support domain is then formed based on the element itself and its adjacent elements sharing common edges (or faces). With the aid of Shepard interpolation, a weighted acoustic gradient field is then formulated, which will be further used to construct the discretized system equations through the generalized Galerkin weak form. Numerical examples demonstrate that the present algorithm can significantly reduces the dispersion error in computational acoustics. Copyright © 2016 John Wiley & Sons, Ltd.     

Generalization of some hypergeometric functions

In this paper, we make use of the Lagrange interpolation to obtain some algebraic identities, involving one or two infinite sets of variables.     

A novel method to calculate the pressure interaction between dust and fluid using SPH

Unstable behavior of smoothed particle hydrodynamics (SPH) dust particles, such as clumping or fingering under certain conditions, has been reported by several researchers who have conducted studies on dusty fluid SPH. The simulation results in this study show that this instability is numerical, and the instability is mainly attributable to the ill‐interpolated pressure gradient in the interaction term between 2 phases. In this paper, we introduce a new method to calculate the pressure force interaction term between dust and fluid particles. The key idea is to first interpolate the pressure gradient at SPH fluid particles and then use the values to calculate the pressure gradient at SPH dust particles, in a consecutive manner. To compare the new method with the existing method, we first conducted an interpolation of pressure gradient at hydrostatic equilibrium under gravity to estimate any error. The results show that the new method is more accurate. We then conducted additional numerical tests, namely, dust‐liquid counterflow, sedimentation in a confined tank, and sedimentation in the presence of turbulence. The unphysical unstable behavior of SPH dust particles such as clumping or fingering was significantly reduced in the new method. The results also show that the instability becomes more significant when using the existing method especially for the case when simulating a flow with relatively high concentration of dust or for the case in which inertia dominates the dynamics of dust particles. Especially, in those cases, the existing method should be avoided, and the newly proposed method is highly recommended.