Kirchhoff-Love Plate Theory: First-Order Analysis, Second-Order Analysis, Plate Buckling Analysis, and Vibration Analysis Using the Finite Difference Method

This paper presents an approach to the Kirchhoff-Love plate theory (KLPT) using the finite difference method (FDM). The KLPT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. Generally in the case of KLPT, the finite difference approximations are derived based on the fourth-order polynomial hypothesis (FOPH) and second-order polynomial hypothesis (SOPH) for the deflection surface. The FOPH is made for the fourth and third derivative of the deflection surface while the SOPH is made for its second and first derivative; this leads to a 13-point stencil for the governing equation. In addition, the boundary conditions and not the governing equations are applied at the plate edges. In this paper, the FOPH was made for all of the derivatives of the deflection surface; this led to a 25-point stencil for the governing equation. Furthermore, additional nodes were introduced at plate edges and at positions of discontinuity (continuous supports/hinges, incorporated beams, stiffeners, brutal change of stiffness, etc.), the number of additional nodes corresponding to the number of boundary conditions at the node of interest. The introduction of additional nodes allowed us to apply the governing equations at the plate edges and to satisfy the boundary and continuity conditions. First-order analysis, second-order analysis, buckling analysis, and vibration analysis of plates were conducted with this model. Moreover, plates of varying thickness and plates with stiffeners were analyzed. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. In first-order, second-order, buckling, and vibration analyses of rectangular plates, the results obtained in this paper were in good agreement with those of well-established methods, and the accuracy was increased through a grid refinement.

Adaptive 27-point finite difference frequency domain method for wave simulation of 3D acoustic wave equation

The finite-difference frequency-domain (FDFD) method has important applications in the wave simulation of various wave equations. To promote the accuracy and efficiency for wave simulation with the FDFD method, we have developed a new 27-point FDFD stencil for 3D acoustic wave equation. In the developed stencil, the FDFD coefficients not only depend on the ratios of cell sizes in the x-, y-, and z-directions, but we also depend on the spatial sampling density (SD) in terms of the number of wavelengths per grid. The corresponding FDFD coefficients can be determined efficiently by making use of the plane-wave expression and the lookup table technique. We also develop a new way for designing an adaptive FDFD stencil by directly adding some correction terms to the conventional second-order FDFD stencil, which is simpler to use and easier to generalize. Corresponding dispersion analysis indicates that, compared to the optimal 27-point stencil derived from the average-derivative method (ADM), the developed adaptive 27-point stencil can reduce the required SD from approximately 4 to 2.2 points per wavelength (PPW) for a cubic mesh and to 2.7 PPW for a general cuboid mesh. Numerical examples of a 3D homogeneous model and SEG/EAGE salt-dome model indicate that the developed stencil is more accurate than the ADM 27-point stencil for cubic and general cuboid meshes, while requiring similar CPU time and computational memory as the ADM 27-point stencil for direct and iterative solvers.

Comparison of High-performance Computing Approaches in the Python Environment for a Five-point Stencil Test Problem

Several of the most important high-performance computing approaches available in the Python programming environment of the LNCC Santos Dumont supercomputer, are compared using a specific test problem. Python includes specific libraries, implementations, development tools, documentation, optimization and parallelization resources. It provides a straightforward way to program using a high level of abstraction, but the parallelization features for exploring multiple cores, processors, or accelerators such as GPUs, are diverse and may not be easily chosen by the user. Serial and parallel implementations of a test problem in Fortran 90 are taken as benchmarks to compare performance. This work is a primer for the use of HPC resources in Python.

New communication strategies in NEMO

<p>One of the main bottlenecks for NEMO scalability is the time spent performing communications. Two complementary strategies are here proposed to reduce the communication frequency and the communication time: the MPI3 neighbourhood collective communications instead of multiple point to point exchanges and the increasing of the halo region size.</p><p>NEMO performs Lateral Boundaries Conditions update by using four point to point MPI communications at north, south, east and west for each MPI domain. The model completes east-west exchange before performing north-south communications. The order of the exchanges allows us to preserve both 5-point and 9-point stencils. MPI3 neighbourhood collectives provide a way to have sub-communicators used to perform collective communications. Two different sub-communicators can be defined in order to support the two different stencils. A single MPI message is needed to be built for all neighbours instead of 4 different messages before calling the collective communication, while the received message is used to update the halo region, following the order of the neighbours in the sub-communicator.</p><p>The new communication strategy has been tested on two computational kernels (i.e. one for 5-point stencil and one for 9-point stencil), selected among the main relevant routines from the computational point of view. Preliminary tests, performed on a domain size of 3000x2000x31 grid points on the Zeus Intel Xeon Gold 6154 machine, available at CMCC, show a gain in communication time for the 5-point stencil use case up to 31% on 2016 cores. The improvement is reduced when communications with processes on the diagonal are activated. However, a modest gain is still achieved, depending on the number of cores.</p><p>On the other side, the analysis of some NEMO routines shows how the exchange of more than one row/column of halo would allow to move communications outside the routine, preserving data dependencies. A wider halo size reduces the frequency of message exchanges whilst increases the message size at each exchange. It allows us to adopt some optimisation strategies (i.e. loop fusion, tiling, etc.) to improve the data locality. Nevertheless, the use of a wider halo introduces itself some improvements for some kernels like for the MUSCL advection scheme which shows a gain of ~23% in the execution time comparing the original version and the new one with halo extended to 2 lines and the communication moved outside the computing region.</p><p>The current work has been performed according to the NEMO development strategy plan, defined by the NEMO Consortium, which establish the priorities of the design strategies to reduce the bottlenecks to the scalability and the time to solution.</p><p> </p><p>Acknowledgments</p><p>This work is co-funded by the EU H2020 IS-ENES project Phase 3 (ISENES3) under Grant Agreement number 824084.</p>

A symmetrical WENO-Z scheme for solving Hamilton–Jacobi equations

In this paper, a new WENO procedure is proposed to approximate the viscosity solution of the Hamilton–Jacobi (HJ) equations. In the one-dimensional (1D) case, an optimum polynomial on a six-point stencil is obtained. This optimum polynomial is fifth-order accurate in regions of smoothness. Then, this optimum polynomial is considered as a symmetric and convex combination of four polynomials with ideal weights. Following the methodology of the classic WENO-Z procedure [Borges et al., J. Comput. Phys. 227, 3191 (2008)], the new nonoscillatory weights are calculated with the ideal weights. Several numerical experiments in 1D, 2D and 3D are performed to illustrate the capability of the scheme.

An improved method for obtaining rotational accelerations from instrumented headforms

The following compares the effect of differentiation methods used to acquire angular acceleration from three types of un-helmeted headform impact tests. The differentiation methods considered were the commonly used 5-point stencil method and a total variation regularization method. Both methods were used to obtain angular acceleration by differentiating angular velocity measured by three angular rate sensors (gyroscopes), and a reference angular acceleration signal was obtained from an array of nine linear accelerometers (that do not require differentiation to obtain angular acceleration). For each impact, three injury criteria that use angular acceleration as an input were calculated from the three angular acceleration signals. The effect of the differentiation methods were considered by comparing the criteria values obtained from gyroscope data to those obtained from the reference signal. Agreement with reference values was observed to be greater for the TV method when a user-defined tuning parameter was optimized for the impact test and cutoff frequency of each condition, particularly at higher cutoff frequencies. In this case, mean absolute error of the five-point stencil ranged from 1.0 (the same) to 11.4 times larger than that associated with the TV method. When a constant tuning parameter value was used across all impacts and cutoff frequencies considered in this study, the TV method still provided a significant improvement over the 5-point stencil method, achieving mean absolute errors as low as one-tenth that observed for the five-point stencil method.