Strong-form Method Research Articles

Electromechanical Piezoelectric Power Harvester Frequency Response Modeling Using Closed-Form Boundary Value Methods

The conversion of mechanical vibration to electrical energy has shown great promise for extending battery life of smart sensor wireless devices for various engineering applications. This paper presents novel analytical models of a piezoelectric bimorph, using the closed-form boundary value (CFBV) method, for predicting the electromechanical power harvester frequency response. The derivations of the coupled electromechanical dynamic response of the transverse-longitudinal (CEDRTL) form based on the CFBV method were developed using the reduced strong form method of the Hamiltonian principle. The equations from CEDRTL can be reduced to give the coupled electromechanical dynamic response of the transverse (CEDRT) form. The electromechanical frequency response functions with variable load resistance were also given in detail using Laplace transformation. The two theoretical studies are compared together and validated with an experimental study. For some cases, when the load resistance approached open circuit, the difference between CEDRTL and CEDRT tended to be more pronounced. Conversely, the CEDRTL and CEDRT models tended to overlap when the load resistance approached short circuit. Nyquist plots are used to demonstrate the shifting frequency and amplitude changes due to variable resistance. Overall, the experimental and CEDRTL model results were very close to each other.

Compact RBF meshless methods for photonic crystal modelling

Meshless methods based on compact radial basis functions (RBFs) are proposed for modelling photonic crystals (PhCs). When modelling two-dimensional PhCs two generalised eigenvalue problems are formed, one for the transverse-electric (TE) mode and the other for the transverse-magnetic (TM) mode. Conventionally, the Band Diagrams for two-dimensional PhCs are calculated by either the plane wave expansion method (PWEM) or the finite element method (FEM). Here, the eigenvalue equations for the two-dimensional PhCs are solved using RBFs based meshless methods. For the TM mode a meshless local strong form method (RBF collocation) is used, while for the tricker TE mode a meshless local weak form method (RBF Galerkin) is used (so that the discontinuity of the dielectric function ϵ ( x ) can naturally be modelled). The results obtained from the meshless methods are found to be in good agreement with the standard PWEM. Thus, the meshless methods are proved to be a promising scheme for predicting photonic band gaps.

A gradient smoothing method (GSM) based on strong form governing equation for adaptive analysis of solid mechanics problems

A gradient smoothing method (GSM) based on strong form of governing equations for solid mechanics problems is proposed in this paper, in which gradient smoothing technique is used successively over the relevant gradient smoothing domains to develop the first- and second-order derivative approximations by calculating weights for a set of field nodes surrounding a node of interest. The GSM is found very stable and can be easily applied to arbitrarily irregular triangular meshes for complex geometry. Unlike other strong form methods, the present method has excellent stability that is crucial for adaptive analysis. An effective and robust residual based error indicator and simple refinement procedure using Delaunay diagram are then implemented in our GSM for adaptive analyses. The reliability and performance of the proposed GSM for adaptive procedure are demonstrated in several solid mechanics problems including problems with singularities and concentrated loading, compared with the well-known finite element method (FEM).

A least-square radial point collocation method for adaptive analysis in linear elasticity

This paper presents a least-square radial point collocation method (LS-RPCM) that is formulated based on the strong formulation and the local approximation using radial basis functions (RBFs). Aiming to solve the instability problem observed in the conventional RPCM using local nodes, a simple and yet effective procedure that uses the well-known least-square technique in a carefully designed manner has been proposed to restore the stability. Since stable solution can now be obtained, the LS-RPCM is then extended for adaptive analysis. Attractive features of the meshfree strong-form method that facilitate the implementation of adaptive analysis are demonstrated via a number of examples in this work. A robust residual based error estimator and a simple refinement procedure using Delaunay diagram are adopted in our adaptive scheme. Stable and accurate results are obtained in all the numerical examples.

A regularized least-squares radial point collocation method (RLS-RPCM) for adaptive analysis

This paper presents a stabilized meshfree method formulated based on the strong formulation and local approximation using radial basis functions (RBFs). The purpose of this paper is two folds. First, a regularization procedure is developed for stabilizing the solution of the radial point collocation method (RPCM). Second, an adaptive scheme using the stabilized RPCM and residual based error indicator is established. It has been shown in this paper that the features of the meshfree strong-form method can facilitated an easier implementation of adaptive analysis. A new error indicator based on the residual is devised and used in this work. As shown in the numerical examples, the new error indicator can reflect the quality of the local approximation and the global accuracy of the solution. A number of examples have been presented to demonstrate the effectiveness of the present method for adaptive analysis.