Spaces of the Haar Type on Arbitrary Irregular Grids

The paper deals with Haar-type spaces on arbitrary irregular grids. The choice of non-uniform grids determines the characteristics of the Haar-type space that can be used to construct the wavelet decomposition. Thus, it becomes a possible adaptive choice of the design space depending on the incoming flow. In contrast to the classical approach, this paper considers the possibility of the adaptive compression of the initial flow. The complexity of the algorithm is directly proportional to the length of the initial number flow. Numerical examples are presented.

Mesh-free radial-basis function-generated finite differences and its application in reverse-time migration

Compared with one-way wave equation migration and ray-based migration, reverse time migration (RTM) using the two-way wave propagation information can produce accurate imaging result for complex structures. Its computational accuracy and efficiency are mainly determined by numerical method for wavefield simulation. When using traditional regular grids for seismic modeling, scattering artifacts may occur due to the stepped approximation of layer interfaces and rugged topography. On the other hand, the irregular grids requires complex grid generation algorithm, despite having certain geometric flexibility. Mesh-free RTM can effectively reduce the scattered noise under regular grids and avoid the extra computation in the process of irregular grids generation. For the implementation of mesh-free RTM method, an algorithm with fast generation of node distributions is used to discretize the underground velocity model, and radial-basis function generated finite-difference (RBF-FD) is used to realize the numerical simulation of wave propagation, cross-correlation imaging condition is adopted for imaging. The mesh-free RTM method which has both flexibility of simulation region and abundance of wavefield information, reduces the storage required for reverse time migration, shows the potential of high-accuracy migration in the case of undulating surface and provides more accurate migration imaging results for oil and gas exploration under complex geological conditions.

Positive Definite Advection Transport Algorithm for Conservation Law Equations on Nonuniform Irregular Grids

The multidimensional positive definite advection transport algorithm (MPDATA) is an important numerical method for the computation of atmospheric dynamics. MPDATA is second-order accurate, positive definite, conservative, and computationally efficient. However, the method is problematic in which it results in a loss of precision when computing a nonuniform irregular grid. Furthermore, research revealed two reasons for this problem. On the one hand, numerical discretization of boundary derivatives of the finite-volume method is incompatible with nonuniform meshes (or grids); on the other hand, the up-wind scheme of staggered grids is not applicable to the calculation of irregular grids. We overcome these two problems by using the multipoint Taylor expansion method to obtain a boundary derivative numerical approximation scheme that does not depend on the grid structure. Furthermore, combined with the well-balance central-upwind scheme, a positive definite advection scheme for irregular meshes is proposed. Then, the positivity of the new numerical scheme is analyzed. Finally, the result of this study is verified by numerical simulation.

Method for Calculating Normal Pressure Distribution of High Resolution and Large Contact Area

The exact calculation of contact stresses below the surface is the basis for optimizing load capacity of heavily loaded rolling–sliding contacts. The level of stress is significantly influenced by the normal pressure distribution within the contact area, which occurs as a result of the transferred normal force and the contact geometry. In this paper, a new method for high resolution pressure calculation of large contact areas is presented. By this, measured surface topography can be taken into account. The basis of the calculation method is the half-space theory according to Boussinesq/Love. Instead of regular grids, optimized meshing strategies are applied to influence the calculation efforts for large contact areas. Two objectives are pursued with the targeted meshing strategy: on the one hand, the necessary resolution for measured surface structures can be realized; while on the other hand, the total number of elements is reduced by a coarse grid in the surrounding areas. In this way, rolling–sliding contacts with large contact areas become computable with conventional simulation computers. Using the newly developed “method of combined solutions,” the overall result is finally composed by the combination of section of separate solutions, which are calculated by consecutively shifting the finely meshed segment over the entire contact area. The vital advancement in this procedure is the introduction of irregular grids, through which the cross influences are not neglected and fully regarded for every separate calculation. The presented methodology is verified stepwise in comparison to the Hertzian theory. The influence of irregular grids on the calculation quality is examined in particular. Finally, the calculation approach is applied to a real disk-on-disk rolling contact based on measured surface topography.