Abstract

We present some developments in the formulation of the Particle Finite Element Method (PFEM) for analysis of complex coupled problems in fluid and solid mechanics accounting for fluid-structure interaction and coupled thermal effects. The PFEM uses an updated Lagrangian description to model the motion of nodes (particles) in both the fluid and the structure domains. Nodes are viewed as material points which can freely move and even separate from the main analysis domain representing, for instance, the effect of water drops. A mesh connects the nodes defining the discretized domain where the governing equations are solved as in the standard FEM. The necessary stabilization for dealing with the incompressibility of the fluid is introduced via the finite calculus (FIC) method. An incremental iterative scheme for the solution of the non linear transient coupled fluid-structure problem is described. Extensions of the PFEM to allow for frictional contact conditions at fluid-solid and solid-solid interfaces via mesh generation are described. A simple algorithm to treat erosion in the fluid bed is presented. Examples of application of the PFEM to solve a number of coupled problems such as the effect of large wave on structures, the large motions of floating and submerged bodies, bed erosion situations and melting and dripping of polymers under the effect of fire are given.

Abstract

The expression ‘finite calculus’ refers to the derivation of the governing differential equations in mechanics by invoking balance of fluxes, forces, etc. in a space–time domain of finite size. The governing equations resulting from this approach are different from those of infinitesimal calculus theory and they incorporate new terms which depend on the dimensions of the balance domain. The new governing equations allow the derivation of naturally stabilized numerical schemes using any discretization procedure. The paper discusses the possibilities of the finite calculus method for the finite element solution of convection–diffusion problems with sharp gradients, incompressible fluid flow and incompressible solid mechanics problems and strain localization situations. 

Abstract

The expression ‘finite calculus’ refers to the derivation of the governing differential equations in mechanics by invoking balance of fluxes, forces, etc. in a space–time domain of finite size. The governing equations resulting from this approach are different from those of infinitesimal calculus theory and they incorporate new terms which depend on the dimensions of the balance domain. The new governing equations allow the derivation of naturally stabilized numerical schemes using any discretization procedure. The paper discusses the possibilities of the finite calculus method for the finite element solution of convection–diffusion problems with sharp gradients, incompressible fluid flow and incompressible solid mechanics problems and strain localization situations. 

Abstract

Flexoelectricity is an electromechanical effect coupling polarization to strain gradients. It fundamentally differs from piezoelectricity because of its size-dependence and symmetry. Flexoelectricity is generally perceived as a small effect noticeable only at the nanoscale. Since ferroelectric ceramics have a particularly high flexoelectric coefficient, however, it may play a significant role as piezoelectric transducers shrink to the submicrometer scale. We examine this issue with a continuum model self-consistently treating piezo- and flexoelectricity. We show that in piezoelectric device configurations that induce strain gradients and at small but technologically relevant scales, the electromechanical coupling may be dominated by flexoelectricity. More importantly, depending on the device design flexoelectricity may enhance or reduce the effective piezoelectric effect. Focusing on bimorph configurations, we show that configurations that are equivalent at large scales exhibit dramatically different behavior for thicknesses below 100¿nm for typical piezoelectric materials. Our results suggest flexoelectric-aware designs for small-scale piezoelectric bimorph transducers.

Abstract

Abstract

This paper presents an enhanced Least Squares Support Vector Machine (LS-SVM) approach for meshless and accurate solution of higher-order boundary value problems (BVPs) that commonly arise in structural mechanics, fluid dynamics, and other engineering fields. The discussed method formulates thirdand fourth-order linear and nonlinear ordinary differential equations (ODEs) as data-driven optimization problems, eliminating the need for traditional mesh-based discretization. Leveraging a Radial Basis Function (RBF) kernel and regularization-based control of model complexity, the LS-SVM captures complex solution behaviour while maintaining stability and smoothness. The meshless nature of the model ensures geometry-independence, making it suitable for irregular or multi-point boundary conditions. A comparative analysis with established machine learning techniques, including Ridge Regression (RR), classical SVM, Random Forest (RF), and Extreme Gradient Boosting (XGB), demonstrates the competitive accuracy, robustness, and efficiency of LSSVM. The results highlight its potential as a promising solver for nonlinear and multi-point problems where meshless methods are advantageous. The results highlight its potential as a promising solver for simulation-based workflows in computational mechanics and scientific computing, where adaptability, generalization, and reliability are critical.OPEN ACCESS Received: 25/08/2025 Accepted: 21/10/2025

Abstract

This paper presents an enhanced Least Squares Support Vector Machine (LS-SVM) approach for meshless and accurate solution of higher-order boundary value problems (BVPs) that commonly arise in structural mechanics, fluid dynamics, and other engineering fields. The discussed method formulates thirdand fourth-order linear and nonlinear ordinary differential equations (ODEs) as data-driven optimization problems, eliminating the need for traditional mesh-based discretization. Leveraging a Radial Basis Function (RBF) kernel and regularization-based control of model complexity, the LS-SVM captures complex solution behaviour while maintaining stability and smoothness. The meshless nature of the model ensures geometry-independence, making it suitable for irregular or multi-point boundary conditions. A comparative analysis with established machine learning techniques, including Ridge Regression (RR), classical SVM, Random Forest (RF), and Extreme Gradient Boosting (XGB), demonstrates the competitive accuracy, robustness, and efficiency of LSSVM. The results highlight its potential as a promising solver for nonlinear and multi-point problems where meshless methods are advantageous. The results highlight its potential as a promising solver for simulation-based workflows in computational mechanics and scientific computing, where adaptability, generalization, and reliability are critical.OPEN ACCESS Received: 25/08/2025 Accepted: 21/10/2025 Published: 16/04/2026