We provide optimal parameter estimates and a priori error bounds for symmetric discontinuous Galerkin (DG) discretisations of the second-order indefinite time-harmonic Maxwell equations. More specifically, we consider two variations of symmetric DG methods: the interior penalty DG (IP-DG) method and one that makes use of the local lifting operator in the flux formulation. As a novelty, our parameter estimates and error bounds are $i)$ valid in the pre-asymptotic regime; $ii)$ solely depend on the geometry and the polynomial order; and $iii)$ are free of unspecified constants. Such estimates are particularly important in three-dimensional (3D) simulations because in practice many 3D computations occur in the pre-asymptotic regime. Therefore, it is vital that our numerical experiments that accompany the theoretical results are also in 3D. They are carried out on tetrahedral meshes with high-order ($p = 1, 2, 3, 4$) hierarchic $H(\mathrm{curl})$-conforming polynomial basis functions.

Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations

In the present research, the dynamic responses of the multilayer functionally graded graphene platelets reinforced composite (FG-GPLRC) spherical panels under blast loading are studied. Three-dimensional elasticity theory is employed to derive the governing equations. The distribution of graphene platelets (GPLs) in each layer is uniform and random with a constant weight fraction. GPLs concentration across the panel thickness may be uniform or graded. The layerwise-differential quadrature method (LW-DQM) together with a non-uniform rational B-spline-based multi-step time integration scheme is used to discretize the motion equations. The convergence behavior of the method is examined numerically. Further, to assure its accuracy, the results in the limit cases are compared with those available in the literature. Finally, through the parametric studies, the effects of material and geometrical parameters such as GPLs distribution patterns, GPLs weight fraction and dimension ratios on the transient responses of the FG-GPLRC spherical panels subjected to blast loading are investigated.

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Three-Dimensional Response of Multilayer FG-GPLRC Spherical Panels Under Blast Loading

The work described in this paper is concerned with providing a rational asymptotic analysis for the partial wrinkling bifurcation of a thin elastic hemispherical segment in which the upper rim experiences in-plane circular shearing relative to the other circular edge. The mathematical structure of the associated complex-valued boundary eigenvalue problem is revealed by using the method of matched asymptotic expansions. Our key result is a three-term asymptotic formula for the critical load in terms of a suitable small parameter proportional to the ratio between the thickness and the radius of the shell. Comparisons of this formula with direct numerical simulations provide further insight into the range of validity of the results derived herein. 

https://pure.hud.ac.uk/ws/files/50352953/torsionREVISED.pdf

Singular Perturbations and Torsional Wrinkling in a Truncated Hemispherical Thin Elastic Shell

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In this paper, a dynamic stiffness formulation is reported to analyze the free vibration characteristics of a rotating cross-ply laminated coupled conical–cylindrical–conical shell. In order to ensure the stability of the numerical results, the whole system is firstly divided into substructures, and each substructure is divided into several segments. The theoretical formulation of the rotating shell segment is derived by the first-order shear deformation theory and Hamilton’s principle. The effects of centrifugal force and Coriolis acceleration on the initial hoop tension are considered in the formulation. The dynamic stiffness matrix of the shell segment is derived from the relationship between the state vector and its derivative, where the state vector consists of the resultant force and displacement components of the shell segment. The dynamic stiffness matrix of the whole system is assembled by the displacement coordination relationship. After the number of segments is determined by the convergence analysis, it is compared with the results of the existing literature and finite element software to ensure the accuracy of the dynamic stiffness formulation. Based on this, the effects of rotational speed, geometry and boundary conditions on the structure are studied. 

A dynamic stiffness formulation for the vibration analysis of rotating cross-ply laminated coupled conical–cylindrical–conical shells

By means of combining Flugge's thin shell theory and energy method, a generalized approach to investigate vibration characteristic of truncated spherical shell subjected to various edge constraints is proposed. The truncated spherical shell is devided into different sections along the meridian line, in which the displacement function of truncated spherical shell along meridian and circumferential line are respectively represented by Jacobi polynomials and Fourier series. Various edge constraints can be simulated on the basis of virtual spring stiffness method in the current research. Finally, the solutions can be derived by meand of Ritz method. The dependability and exactness of current method have been proved by the comparison between current method, FEM and related literatures. The dimensionless frequency parameters of different truncated spherical shell under various edge constraints are displayed. In addition, the influence of geometric dimensions and boundary constraints on frequency parameters are also discussed.

Yuhui Li

Papers

Vibration analysis of truncated spherical shells under various edge constraints