An implicit mesh discontinuous Galerkin formulation for higher-order plate theories

Abstract

In this work, a discontinuous Galerkin formulation for higher-order plate theories is presented.The starting point of the formulation is the strong form of the governing equations, which are derived in the context of the Generalized Unified Formulation and the Equivalent Single Layer approach from the Principle of Virtual Displacements. To express the problem within the discontinuous Galerkin framework, an auxiliary flux variable is introduced and the governing equations are rewritten as a system of first-order partial differential equations, which are weakly stated over each mesh element. The link among neighbouring mesh elements is then retrieved by introducing suitably defined numerical fluxes, whose explicit expressions define the proposed Interior Penalty discontinuous Galerkin formulation. Furthermore, to account for the presence of generally curved boundaries of the considered plate domain, the discretisation mesh is built by combining a background grid and an implicit representation of the domain. hp-convergence analyses and a comparison with the results obtained using the Finite Element Method are provided to show the accuracy of the proposed formulation as well as the computational savings in terms of overall degrees of freedom.

Layer-Wise Discontinuous Galerkin Methods for Piezoelectric Laminates

“…Consistently with the notation introduced in [30,31], Equation (8) can be expressed in matrix form as…”

“…The derivation of a solution scheme based on the discontinuous Galerkin method generally involves [23]: (i) the introduction of an auxiliary variable that allows to rewrite the second-order system as a first-order system; and (ii) a weak statement of said first-order system for each element of the mesh discretizing the considered domain. Here, the derivation proposed by Gulizzi et al [30,31] is employed.…”

“…However, prior to giving their explicit expression, it is useful to introduce ∂Ω N as the subsets of ∂Ω (e) where Dirichlet and Neumann boundary conditions, respectively, are to be enforced; furthermore, considering that the mesh discretization introduces a subdivision of the plate domain Ω, we define ∂Ω (e) I as the part of ∂Ω (e) that the element e shares with its neighboring elements. Then, upon employing the interior penalty dG formulation introduced in [30,31], the numerical fluxes are chosen as follows…”

“…where ||U|| 2 is the L 2 (Ω) norm, U ref is the Navier solution and U h is the solution computed using the present formulation. It is worth stressing that the solution fields U h = U h (X 1 , X 2 ) provided by the developed dG scheme and assessed through the error definition in Equation (30) are those appearing in the generalized kinematical model in Equation 9, which identifies them as a generalization of the mid-plane displacements and rotations employed in classical plate theories. Figure 3a shows the effect of the penalty parameter µ introduced in Equation (24) on the error measure e(U h ) as a function of the basis functions order p and for a mesh grid 2 × 2 and a layer-wise theory LW 3 .…”

“…When compared to the FEM literature, the adoption of dG-based numerical schemes for multilayered plates appears quite limited and mostly focused on the Classical Laminated Plate Theory [24][25][26][27] or the First-order Shear Deformation Theory [28,29]. Recently, Gulizzi et al [30,31] presented dG formulations for ESL and LW theories for elastic plates, whereas, to the best of the Authors' knowledge, no discontinuous Galerkin formulation for LW theories of piezoelectric plates is available.…”