On the use of exponential basis functions in the analysis of shear deformable laminated plates

In this report, we introduce a meshfree approach for static analysis of isotropic/orthotropic crossply laminated plates with symmetric/non-symmetric layers. Classical, first and third order shear deformation plate theories are employed to perform the analyses. In this method, the solution is first split into homogenous and particular parts and then the homogenous part is approximated by the summation of an appropriately selected set of exponential basis functions (EBFs) with unknown coefficients. In the homogenous solution the EBFs are restricted to satisfy the governing differential equation. The particular solution is derived using a similar approach and another series of EBFs. The imposition of the boundary conditions and determination of the unknown coefficients are performed by a collocation method through a discrete transformation technique. The solution method allows us to obtain semi-analytical solution of plate problems with various shapes and boundary conditions. The solutions of several benchmark plate problems with various geometries are presented to validate the results

Exponential basis functions in solution of incompressible fluid problems with moving free surfaces

In this report, a new simple meshless method is presented for the solution of incompressible inviscid fluid flow problems with moving boundaries. A Lagrangian formulation established on pressure, as a potential equation, is employed. In this method, the approximate solution is expressed by a linear combination of exponential basis functions (EBFs), with complex-valued exponents, satisfying the governing equation. Constant coefficients of the solution series are evaluated through point collocation on the domain boundaries via a complex discrete transformation technique. The numerical solution is performed in a time marching approach using an implicit algorithm. In each time step, the governing equation is solved at the beginning and the end of the step, with the aid of an intermediate geometry. The use of EBFs helps to find boundary velocities with high accuracy leading to a precise geometry updating. The developed Lagrangian meshless algorithm is applied to variety of linear and nonlinear benchmark problems. Non-linear sloshing fluids in rigid rectangular two-dimensional basins are particularly addressed

HDG-NEFEM with Degree Adaptivity for Stokes Flows

This paper presents the first degree adaptive procedure able to directly use the geometry given by a CAD model. The technique uses a hybridisable discontinuous Galerkin discretisation combined with a NURBS-enhanced rationale, completely removing the uncertainty induced by a polynomial approximation of curved boundaries that is common within an isoparametric approach. The technique is compared against two strategies to perform degree adaptivity currently in use. This paper demonstrates, for the first time, that the most extended technique for degree adaptivity can easily lead to a non-reliable error estimator if no communication with CAD software is introduced whereas if the communication with the CAD is done, it results in a substantial computing time. The proposed technique encapsulates the CAD model in the simulation and is able to produce reliable error estimators irrespectively of the initial mesh used to start the adaptive process. Several numerical examples confirm the findings and demonstrate the superiority of the proposed technique. The paper also proposes a novel idea to test the implementation of high-order solvers where different degrees of approximation are used in different elements