The J-area integral applied in peridynamics

The J-integral is in its original formulation expressed as a contour integral. The contour formulation was, however, found cumbersome early on to apply in the finite element analysis, for which method the more directly applicable J-area integral formulation was later developed. In a previous study, we expressed the J-contour integral as a function of displacements only, to make the integral directly applicable in peridynamics (Stenström and Eriksson in Int J Fract 216:173–183, 2019). In this article we extend the work to include the J-area integral by deriving it as a function of displacements only, to obtain the alternative method of calculating the J-integral in peridynamics as well. The properties of the area formulation are then compared with those of the contour formulation, using an exact analytical solution for an infinite plate with a central crack in Mode I loading. The results show that the J-area integral is less sensitive to local disturbances compared to the contour counterpart. However, peridynamic implementation is straightforward and of similar scope for both formulations. In addition, discretization, effects of boundaries, both crack surfaces and other boundaries, and integration contour corners in peridynamics are considered.

Morrey Spaces Related to Schrödinger Operators with Certain Nonnegative Potentials and Littlewood–Paley–Stein Functions on the Heisenberg groups

Let be a Schrödinger operator on the Heisenberg group , where is the sublaplacian on and the nonnegative potential V belongs to the reverse Hölder class with . Here is the homogeneous dimension of . Assume that is the heat semigroup generated by. The Lusin area integral and the Littlewood–Paley–Stein function associated with the Schrödinger operator are defined, respectively, bywhereandWhere is a parameter. In this article, the author shows that there is a relationship between and the operator and for any , the following inequality holds true:Based on this inequality and known results for the Lusin area integral , the author establishes the strong-type and weak-type estimates for the Littlewood–Paley–Stein function on . In this article, the author also introduces a class of Morrey spaces associated with the Schrödinger operator on . By using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the operator acting on the Morrey spaces for an appropriate choice of . It can be shown that the same conclusions hold for the operator on generalized Morrey spaces as well.

A novel prediction method for rail grinding profile based on an interval segmentation approach and accurate area integral with cubic NURBS

Rail grinding profile prediction in different grinding patterns is important to improve the grinding quality for the rail grinding operation site. However, because of high-dimensional and strong nonlinearity between grinding amount and grinding parameters, the prediction error and computational cost is relatively high. As a result, the accuracy and efficiency of conventional methods cannot be guaranteed. In this article, an accurate and efficient rail grinding profile prediction method is proposed, in which an interval segmentation approach is proposed to improve the prediction efficiency based on the geometric characteristic of a rail profile. Then, the accurate area integral approach with cubic NURBS is used as the grinding area calculation approach to improve the prediction accuracy. Finally, the normal length index is introduced to evaluate the prediction accuracy. The accuracy and stability of the proposed method are verified by comparing a conventional approach based on a practical experiment. The results demonstrate that the proposed method can predict the rail grinding profile in any grinding pattern with high accuracy and efficiency. Meanwhile, its prediction stability basically agrees with the conventional approach.

Area Integral Characterization of Hardy space H1L related to Degenerate Schrödinger Operators

Let $$\begin{array}{} \displaystyle Lf(x)=-\frac{1}{\omega(x)}\sum_{i,j}^{}\partial_{i}(a_{ij}(\cdot)\partial_{j}f)(x)+V(x)f(x) \end{array}$$ be the degenerate Schrödinger operator, where ω is a weight from the Muckenhoupt class A2, V is a nonnegative potential that belongs to a certain reverse Hölder class with respect to the measure ω(x)dx. For such an operator we define the area integral $\begin{array}{} \displaystyle S^{L}_h \end{array}$ associated with the heat semigroup and obtain the area integral characterization of $\begin{array}{} \displaystyle H^{1}_{L} \end{array}$, which is the Hardy space associated with L.