The Meshless Analysis of Scale-Dependent Problems for Coupled Fields

The meshless local Petrov–Galerkin (MLPG) method was developed to analyze 2D problems for flexoelectricity and higher-grade thermoelectricity. Both problems were multiphysical and scale-dependent. The size effect was considered by the strain and electric field gradients in the flexoelectricity, and higher-grade heat flux in the thermoelectricity. The variational principle was applied to derive the governing equations within the higher-grade theory of considered continuous media. The order of derivatives in the governing equations was higher than in their counterparts in classical theory. In the numerical treatment, the coupled governing partial differential equations (PDE) were satisfied in a local weak-form on small fictitious subdomains with a simple test function. Physical fields were approximated by the moving least-squares (MLS) scheme. Applying the spatial approximations in local integral equations and to boundary conditions, a system of algebraic equations was obtained for the nodal unknowns.

Analysis of Piezoelectric Semiconducting Solids by Meshless Method

In this paper, a meshless local Petrov-Galerkin (MLPG) method is proposed to calculate mechanical and electrical responses of three-dimensional piezoelectric semiconductors under static load. The analyzed solid is discretized by a set of generally distributed nodal points distributed over 3D geometry. Local integral equations (LIEs) are derived from the weak form of governing equations over small local subdomains. The subdomains have a spherical shape with a nodal point located in its centre. A unit step function is used as the test functions in the local weak-form. The moving least-squares (MLS) method is adopted for the approximation of the physical quantities in the LIEs. The proposed MLPG method is verified by using the corresponding results obtained with the finite element method. Numerical examples are presented and discussed for various boundary conditions and loading scenarios to show the performance of the developed MLPG method for analysis piezoelectric semiconducting solids.

Longitudinally Invariant Elastic Structures Analyzed by the Meshless Mlpg Method Using 2.5D Approach

This paper presents a general 2.5D meshless MLPG methodology for the computation of the elastic response of longitudinally invariant structure subjected to the incident wave field. A numerical frequency domain model is established using the Fourier transform in time and longitudinal coordinate domains. This allows for significant reduction of computational effort required. In the MLPG method the Moving-Least Squares (MLS) scheme is employed for the approximation of the spatial variation of displacement field. No finite elements are required for the approximation or integration of unknowns. A small circular subdomain is introduced around each nodal point in the analyzed domain. Local integral equations derived from the governing equations are specified on these subdomains. Continuously non-homogeneous material properties are varying in the cross-section of the analyzed structure. A simple patch test is introduced to assess the accuracy and the convergence of developed numerical model. At the end of the paper, numerical examples illustrate the applicability of the proposed numerical formulation.

Crack Analysis in Piezoelectric Semiconductors

Mechanical and electric loads are considered for 2-d crack problems in conducting piezoelectric materials. The electric displacement in conducting piezoelectric materials is influenced by the electron density and it is coupled with the electric current. The coupled governing partial differential equations (PDE) for stresses, electric displacement field and current are satisfied in a local weak-form on small fictitious subdomains. Nodal points are spread on the analyzed domain and each node is surrounded by a small circle for simplicity. Local integral equations are derived for a unit function as the test function on circular subdomains. All field quantities are approximated by the moving least-squares (MLS) scheme.

Numerical MLPG Analysis of Piezoelectric Sensor in Structures

The paper deals with a numerical analysis of the electro-mechanical response of piezoelectric sensors subjected to an external non-uniform displacement field. The meshless method based on the local Petrov-Galerkin (MLPG) approach is utilized for the numerical solution of a boundary value problem for the coupled electro-mechanical fields that characterize the piezoelectric material. The sensor is modeled as a 3-D piezoelectric solid. The transient effects are not considered. Using the present MLPG approach, the assumed solid of the cylindrical shape is discretized with nodal points only, and a small spherical subdomain is introduced around each nodal point. Local integral equations constructed from the weak form of governing PDEs are defined over these local subdomains. A moving least-squares (MLS) approximation scheme is used to approximate the spatial variations of the unknown field variables, and the Heaviside unit step function is used as a test function. The electric field induced on the sensor is studied in a numerical example for two loading scenarios.