Stabilization for Equal-Order Polygonal Finite Element Method for High Fluid Velocity and Pressure Gradient

This paper presents an adapted stabilisation method for the equal-order mixed scheme of finite elements on convex polygonal meshes to analyse the high velocity and pressure gradient of incompressible fluid flows that are governed by Stokes equations system.This technique is constructed by a local pressure projection which is extremely simple,yet effective,to eliminate the poor or even non-convergence as well as the instability of equal-order mixed polygonal technique.In this research,some numerical examples of incompressible Stokes fluid flow that is coded and programmed by MATLAB will be presented to examine the effectiveness of the proposed stabilised method.

Polygonal Finite Element for Two-Dimensional Lid-Driven Cavity Flow

This paper investigates a polygonal finite element(PFE)to solve a two-dimensional(2D)incompressible steady fluid problem in a cavity square.It is a well-known standard benchmark(i.e.,lid-driven cavity flow)-to evaluate the numerical methods in solving fluid problems controlled by the Navier-Stokes(N-S)equation system.The approximation solutions provided in this research are based on our developed equal-order mixed PFE,called Pe1Pe1.It is an exciting development based on constructing the mixed scheme method of two equal-order discretisation spaces for both fluid pressure and velocity fields of flows and our proposed stabilisation technique.In this research,to handle the nonlinear problem of N-S,the Picard iteration scheme is applied.Our proposed method’s performance and convergence are validated by several simulations coded by commercial software,i.e.,MATLAB.For this research,the benchmark is executed with variousReynolds numbers up to the maximum Re=1000.All results then numerously compared to available sources in the literature.

A novel virtual node method for polygonal elements

A novel polygonal finite element method （PFEM） based on partition of unity is proposed, termed the virtual node method （VNM）. To test the performance of the present method, numerical examples are given for solid mechanics problems. With a polynomial form, the VNM achieves better results than those of traditional PFEMs, including the Wachspress method and the mean value method in standard patch tests. Compared with the standard triangular FEM, the VNM can achieve better accuracy. With the ability to construct shape functions on polygonal elements, the VNM provides greater flexibility in mesh generation. Therefore, several fracture problems are studied to demonstrate the potential implementation. With the advantage of the VNM, the convenient refinement and remeshing strategy are applied.

A Quadratic Serendipity Finite Volume Element Method on Arbitrary Convex Polygonal Meshes

Based on the idea of serendipity element,we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonalmeshes in this article.The explicit construction of quadratic serendipity element shape function is introduced from the linear generalized barycentric coordinates,and the quadratic serendipity element function space based on Wachspress coordinate is selected as the trial function space.Moreover,we construct a family of unified dual partitions for arbitrary convex polygonal meshes,which is crucial to finite volume element scheme,and propose a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom.Finally,under certain geometric assumption conditions,the optimal H1 error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained,and verified by numerical experiments.

Adaptive phase field modelling of crack propagation in orthotropic functionally graded materials

In this work,we extend the recently proposed adaptive phase field method to model fracture in orthotropic functionally graded materials(FGMs).A recovery type error indicator combined with quadtree decomposition is employed for adaptive mesh refinement.The proposed approach is capable of capturing the fracture process with a localized mesh refinement that provides notable gains in computational efficiency.The implementation is validated against experimental data and other numerical experiments on orthotropic materials with different material orientations.The results reveal an increase in the stiffness and the maximum force with increasing material orientation angle.The study is then extended to the analysis of orthotropic FGMs.It is observed that,if the gradation in fracture properties is neglected,the material gradient plays a secondary role,with the fracture behaviour being dominated by the orthotropy of the material.However,when the toughness increases along the crack propagation path,a substantial gain in fracture resistance is observed.