Polygonal Finite Element Method Research Articles

Fatigue crack growth in functionally graded materials using an adaptive phase field method with cycle jump scheme

Functionally graded materials provide versatility in adjusting the volume fractions of constituent materials to meet specific design requirements. However, this customization often introduces mode-mixity at the crack tip, posing challenges in predicting fracture under cyclic loading with discrete approaches and computationally expensive with conventional phase-field fracture models. To address these issues, this paper introduces an adaptive phase-field fracture formulation with cycle jump scheme to elegantly predict fatigue crack nucleation and growth in functionally graded materials. Within this framework, the effective properties at a point are estimated using the Mori–Tanaka homogenization scheme, while the crack growth due to cyclic load is captured by incorporating an additional fatigue degradation parameter. Moreover, the computational efficiency of the proposed framework is improved through an adaptive mesh refinement and explicit cycle jump scheme. The adaptive refinement scheme utilizes an error indicator derived from both the displacement solution and phase-field variable. The adaptive refinement scheme is integrated with efficient quadtree decomposition, which generates a hierarchical mesh structure. Hanging nodes resulting from the quadtree decomposition are efficiently handled using a polygonal finite element method. The proposed framework is validated against experimental and numerical results reported in the literature. Furthermore, we investigate the fatigue crack growth resistance across a broad range of material gradation directions, gaining valuable insights and identifying functionally graded materials with high fatigue resistance.

An n-sided polygonal cell-node-based smoothed finite element method for solving two-dimensional heat conduction problems

An n-sided polygonal cell-node-based smoothed finite element method is proposed to analyze two-dimensional heat conduction problems. Through the gradient smoothing technique, the internal integral of the polygonal element is transformed into the boundary integral based on the smoothing domain, thus reducing the continuity requirement of the trial function, and only requiring the shape function value of the smoothing domain boundary calculated using the Wachspress coordinates. Therefore, when constructing the smoothed finite element form of the heat conduction problems, coordinate mapping is avoided, which greatly improves the computational efficiency. Furthermore, a new division of the cell-based smoothing domain is put forward to improve the accuracy of the solution, where the divisions of the node-based and cell-based smoothing domains are combined in background elements to form cell-node-based smoothing domains. Extensive numerical experiments are carried out to show that the current model performs better in the estimation of the accuracy of the solution, the computational cost, the equivalent energy, and the temperature gradient while maintaining the same precision as the polygonal finite element method (PFEM).

A polygonal finite element method for shakedown analysis of structures

A polygonal finite element method for shakedown analysis of structures

The polygonal finite element method for solving heat conduction problems

In this paper, a general polygonal finite element method (FEM-polygon) based on the Wachspress shape function is used to solve two-dimensional heat conduction problems. The finite element variational form of the heat conduction problem is first established based on the weighted residual method. Then the formulations of the thermal stiffness matrix, thermal damping matrix, and nodal heat vector are derived using the Wachspress coordinates. The global stiffness matrix is further assembled and the well-behaved algebraic equations of the thermal system are established. Finally, intensive numerical tests are conducted to verify the super convergence in temperature and the high precision in equivalent energy and temperature gradient of the present FEM-polygon for 2D heat conduction problems. Besides, the proposed model also performs excellently for traditional triangular and quadrilateral elements.

Polygonal finite element-based content-aware image warping

Mesh-based image warping techniques typically represent image deformation using linear functions on triangular meshes or bilinear functions on rectangular meshes. This enables simple and efficient implementation, but in turn, restricts the representation capability of the deformation, often leading to unsatisfactory warping results. We present a novel, flexible polygonal finite element (poly-FEM) method for content-aware image warping. Image deformation is represented by high-order poly-FEMs on a content-aware polygonal mesh with a cell distribution adapted to saliency information in the source image. This allows highly adaptive meshes and smoother warping with fewer degrees of freedom, thus significantly extending the flexibility and capability of the warping representation. Benefiting from the continuous formulation of image deformation, our poly-FEM warping method is able to compute the optimal image deformation by minimizing existing or even newly designed warping energies consisting of penalty terms for specific transformations. We demonstrate the versatility of the proposed poly-FEM warping method in representing different deformations and its superiority by comparing it to other existing state-of-the-art methods.

A Novel Least-Squares Level Set Method by Using Polygonal Elements

In this study, we apply an artificial viscosity method to convert an unsteady level set (LS) convection equation into an unsteady LS convection-diffusion transport equation to stabilize the numerical solution of the convection term. Then a novel least-square polygonal finite element method is used to solve an unsteady LS convection-diffusion problem. The least-squares method provided good mathematical properties such as natural numerical diffusion and the positive definite symmetry of the resulting algebraic systems for the convection-diffusion and re-initialization equations. The proposed method is evaluated numerically in two different benchmark problems: a rigid body rotation of Zalesak’s disk, and a time-reversed single-vortex flow. In comparison with conventional triangular (T3) and quadrilateral (Q4) elements, polygonal elements are capable of providing greater flexibility in mesh generation for complicated problems as well as more accurate in solving the LS equations. In addition, the numerical results are also compared with the results which obtained from essentially non-oscillatory type formulations and particle LS methods. The results show that the proposed method completely matches the previously published results.

Communication—A Fast and Accurate Numerical Technique for Impedance Spectroscopy of Microstructures

The polygonal finite element method (PFEM) is proposed as a fast and accurate technique to simulate the impedance spectroscopy (IS) of polycrystalline materials. While conventional finite element method (FEM) requires explicit meshing of the grains and grain boundaries, in PFEM each region can be treated as an element. We demonstrate that the number of degrees of freedom in PFEM can be lower by a factor of 30 when compared to FEM, thus speeding up simulations by a factor of 3.5. A simple example demonstrates the use of PFEM to generate IS on samples with various grain boundary widths.

A cell-based smoothed finite element method for finite elasticity

In this study, we present a displacement based polygonal finite element method for compressible and nearly-incompressible elastic solids undergoing large deformations in two dimensions. This is achieved by projecting the dilatation strain onto the linear approximation space, within the framework of volume averaged nodal projection method. To reduce the numerical integration burden over polytopes, a linear strain smoothing technique is employed to compute the terms in the bilinear/linear form. The salient features of the proposed framework are: (a) does not require derivatives of shape functions and complex numerical integration scheme to compute the bilinear and linear form and (b) volumetric locking is alleviated by adopting the volume averaged nodal projection technique. The efficacy, convergence properties and accuracy of the proposed framework is demonstrated through four standard benchmark problems.

A polygonal FEM and continuum damage mechanics based framework for stochastic simulation of fatigue life scatter in duplex microstructure titanium alloys

The scatter in the fatigue behavior of duplex microstructure titanium alloys occurs due to microstructural randomness. The presence of two phases in titanium alloy microstructure also contributes to the scatter in fatigue life data. In the literature, several mathematical frameworks have been developed to consider the effect of microstructural randomness on the prediction of fatigue scatter. However, no numerical study has been found that considers the effect of presence of two phases on the fatigue life scatter. Thus, in the present work, a mathematical framework based on continuum damage mechanics (CDM) and polygonal finite element method (PFEM) is developed to predict the fatigue life scatter in α+β titanium alloys. In this framework, PFEM is applied to perform the numerical simulations and CDM is used to explain the fatigue behavior of material. In PFEM, the Voronoi tessellation approach is used to generate the idealized microstructure models. The Waschpress shape functions are adopted to compute the shape functions over the arbitrary convex polygonal elements. In CDM, a fatigue damage model is adopted to consider the crack initiation and propagation under the cyclic loading conditions. The advantage of combined CDM-PFEM framework is that the numerical simulation can be performed at relatively coarser mesh which decreases the computational cost and increases the efficiency of the computational model.While predicting the fatigue life, it has been observed that micro cracks may initiate and propagate at several locations inside the microstructure. Thus, to track the nucleation and growth of micro cracks effectively, a machine learning based technique ‘cluster analysis’ is employed in the developed framework. The “cluster analysis” identifies the location and dimensions of the micro and dominant crack patterns in the domain. Furthermore, for an idealized representation of duplex microstructures, both phases of titanium alloy (i.e., primary α and β) are modeled using random distribution function. The effect of different sources of randomness in microstructures i.e., topology of microstructure, grain size, volume fraction of alpha phase, inhomogeneity in elastic modulus and internal voids, on the fatigue life is investigated. Several simulations are performed on the numerically generated microstructures at various loads. The statistical analysis of the predicted fatigue lives data is performed through normal and Weibull distribution fits. The simulation results highlight that the combined CDM-PFEM based framework is simple and an effective tool to predict the fatigue life scatter and can effectively analyze the mean fatigue life in titanium alloys under cyclic loading conditions.

A rate-dependent constitutive model incorporated in two-dimensional PolyFEM for Galfenol sensors

A micromechanical model with rate effects is developed to constitute the nonlinear hysteresis sensor behaviour of Galfenol and embedded into a numerical framework, the polygonal finite element method. The model stems from the domain rotation events taking place at microlevel in response to the external magnetomechanical loads. The internal variables of the model to represent the material state at any instant are the volume fraction of the domains, and their evolution equations are derived using thermodynamic principles. Inter-domain effects within each grain are accounted in the form of back fields that act as hardening or softening parameters. The constitutive model is extended further to include the rate effects and then incorporated into a hybrid-magnetomechanical-formulation with Voronoi-based-discretization to study the behaviour of Galfenol. The Voronoi discretizations closely resemble the microstructure of Galfenol with each finite element specifying a random grain in the polycrystalline Galfenol. The developed framework is shown to simulate the major and minor loop sensor responses of the material with rate effects under complex loading conditions.

Error estimation for the polygonal finite element method for smooth and singular linear elasticity

A recovery-based error indicator developed to evaluate the quality of polygonal finite element approximations is presented in this paper. Generalisations of the finite element method to arbitrary polygonal meshes have been increasingly investigated in the last years, as they provide flexibility in meshing and improve solution accuracy. As any numerical approximation, they have an induced error which has to be accounted for in order to validate the approximate solution. Here, we propose a recovery type error measure based on a moving least squares fitting of the finite element stress field. The quality of the recovered field is improved by imposing equilibrium conditions and, for singular problems, splitting the stress field into smooth and singular parts. We assess the performance of the error indicator using three problems with exact solution, and we also compared the results with those obtained with standard finite element meshes based on simplexes. The results indicate good values for the local and global effectivities, similar to the values obtained for standard approximations, and are always within the recommended range.

Microstructure‐related modeling of timber constructions

AbstractWood is an organic porous material of cellulose fibers, which consist of a cell wall and a lumen. In this paper a microstructure‐related model utilizing the polygonal finite element method (PFEM) is presented. The PFEM enables the discrete modeling of the wooden cellular structure under application of shape functions based on mean value coordinates. The numerical integration scheme is based on the triangulation of the elements. In the scope of this framework the transverse compression behavior of wood is simulated. The deformation is presented and compared to the macrostructure‐related result.

A novel computational approach to functionally graded porous plates with graphene platelets reinforcement

Abstract In this study, we numerically investigate static and free vibration responses of functionally graded (FG) porous plates with graphene platelets (GPLs) reinforcement using an efficient polygonal finite element method (PFEM). While the bending strain field is approximated through quadratic serendipity shape functions, the shear strain field is calculated by employing Wachspress basis functions. In order to eliminate the shear locking phenomenon, Timoshenko's beam theory is utilized to determine assumed strain fields on each side of polygonal domain. The present formulation possesses various outstanding features: (a) is valid for triangular, quadrilateral and polygonal elements; (b) can conveniently implement various different plate theories via choosing appropriate transverse shear function; (c) eliminates the shear locking phenomenon; (d) does not increase degrees of freedom (DOFs) per polygonal element despite employing the quadratic serendipity shape functions and (e) obtains more accurate and stable results than those of other PFEMs. Various dispersions of internal pores as well as GPLs into metal matrix through the thickness of plate are examined. The effective material properties varying across the plate's thickness can be estimated by Halpin-Tsai model for Young's modulus and the rule of a mixture for Poisson's ratio and mass density. The effect of several important parameters such as porosity coefficient, weight fraction and dimensions of GPLs, distribution of porosity and GPLs into metal matrix are thoroughly investigated via various numerical examples.

On the application of polygonal finite element method for Stokes flow – A comparison between equal order and different order approximation

In this work, we discuss the application of polygonal finite elements to the solution of Stokes equations in two dimensions. The proposed framework is based on the mixed finite element discretization that involves velocity and pressure as independent unknown field variables. We present two approaches: (a) same order of approximation for velocity and pressure with Pressure Stabilization Petrov-Galerkin method (PSPG) and (b) different order of approximation for velocity and pressure. The approximation functions over polygons are rational polynomials based on Wachspress coordinates and for numerical integration of the terms in the bilinear and the linear form, we employ the linear smoothing technique. The relative performance between the approaches, the convergence properties and the accuracy is presented for two dimensional numerical examples, which shows that the proposed framework yields accurate results and converges at optimal convergence rate.

Adaptive phase field method for quasi-static brittle fracture using a recovery based error indicator and quadtree decomposition

An adaptive phase field method is proposed for crack propagation in brittle materials under quasi-static loading. The adaptive refinement is based on the recovery type error indicator, which is combined with the quadtree decomposition. Such a decomposition leads to elements with hanging nodes. Thanks to the polygonal finite element method, the elements with hanging nodes are treated as polygonal elements and hence do not require any special treatment. The mean value coordinates are used to approximate the unknown field variables and a staggered solution scheme is adopted to compute the displacement and the evolution of the phase field variable. A few standard benchmark problems are solved to show the efficiency of the proposed framework. It is seen that the proposed framework yields comparable results at a fraction of the computational cost when compared to standard approaches reported in the literature.

A high-order mixed polygonal finite element for incompressible Stokes flow analysis

This paper presents a new high-order mixed scheme of polygonal finite element method (PFEM) for incompressible flow computation. The system of polygonal basis functions with bubble-enriched functions is applied to represent the fluid velocity field, while the pressure field is approximated by the set of polygonal basis functions. The new element is named as MINIPe. From a practical point of view, the present element naturally satisfies the well-known inf–sup condition. The numerical tests of the incompressible fluid flow problems governed by Stokes equation systems are presented to verify the performance quality of proposed method with respect to the stability and convergence rate.

Numerical modeling of rock failure under dynamic loading with polygonal elements

SummaryPolygonal finite elements are gaining an increasing attention in the computational mechanics literature, but their application in rock mechanics is very rare. This paper deals with numerical modeling of rock failure under dynamic loading based on polygonal finite elements. For this end, a damage‐viscoplastic constitutive model for rock based on the Mohr‐Coulomb criterion with the Rankine criterion as a tensile cutoff is employed and implemented with the polygonal finite element method. Moreover, the mineral mesostructure or rock is described by randomly mapping groups of polygonal elements representing the constituent minerals into a global mesh and assigning these groups with the corresponding mineral material properties. The performance of the polygonal elements is compared with that of the linear and quadratic triangular and bilinear quadrilateral elements in numerical simulations of controlled shear band formation under uniaxial compression and lateral splitting failure in the dogbone tension test. Numerical simulations of uniaxial tension and compression tests as well as dynamic Brazilian disc test under increasing loading rates demonstrate that the present approach predicts the correct failure modes as well as the dynamic increase in strength of rock.

An extended polygonal finite element method for large deformation fracture analysis

The modeling of large deformation fracture mechanics has been a challenging problem regarding the accuracy of numerical methods and their ability to deal with considerable changes in deformations of meshes where having the presence of cracks. This paper further investigates the extended finite element method (XFEM) for the simulation of large strain fracture for hyper-elastic materials, in particular rubber ones. A crucial idea is to use a polygonal mesh to represent space of the present numerical technique in advance, and then a local refinement of structured meshes at the vicinity of the discontinuities is additionally established. Due to differences in the size and type of elements at the boundaries of those two regions, hanging nodes produced in the modified mesh are considered as normal nodes in an arbitrarily polygonal element. Conforming these special elements becomes straightforward by the flexible use of basis functions over polygonal elements. Results of this study are shown through several numerical examples to prove its efficiency and accuracy through comparison with former achievements.

Geometrically nonlinear polygonal finite element analysis of functionally graded porous plates

In this study, an efficient polygonal finite element method (PFEM) in combination with quadratic serendipity shape functions is proposed to study nonlinear static and dynamic responses of functionally graded (FG) plates with porosities. Two different porosity types including even and uneven distributions through the plate thickness are considered. The quadratic serendipity shape functions over arbitrary polygonal elements including triangular and quadrilateral ones, which are constructed based on a pairwise product of linear shape functions, are employed to interpolate the bending strains. Meanwhile, the shear strains are defined according to the Wachspress coordinates. By using the Timoshenko's beam to interpolate the assumption of the strain field along the edges of polygonal element, the shear locking phenomenon can be naturally eliminated. Furthermore, the C0–type higher-order shear deformation theory (C0–HSDT), in which two additional variables are included in the displacement field, significantly improves the accuracy of numerical results. The nonlinear equations of static and dynamic problems are solved by Newton–Raphson iterative procedure and by Newmark's integration scheme in association with the Picard methods, respectively. Through various numerical examples in which complex geometries and different boundary conditions are involved, the proposed approach yields more stable and accurate results than those generated using other existing approaches.

A polytree-based adaptive polygonal finite element method for topology optimization of fluid-submerged breakwater interaction

The polytree-based adaptive polygonal finite element method was recently proved to be very efficient for a wide class of mechanics problems. In this study, we further exploit its advantages for topology optimization of fluid-submerged breakwater interaction. Due to the inherent limitations of triangular and quadrilateral elements for topology optimization, an alternative formulation of using arbitrary polygons based on the Wachspress coordinate is employed for structural domain. In addition, we use a polytree-based adaptive technique to enhance the accuracy for topology problem in structural domain with a minimum number of degrees of freedom (DOF). We apply this method to fluid–structure interaction (FSI) problems and then finding an optimal shape of submerged breakwater (SBW), which is as a special kind of coastal structure located in front of the beach and underwater to protect the landside from the erosion. Several numerical results demonstrate the effectiveness of the present method. It is observed that the obtained results are in good agreement with reference solutions.