Polygonal Elements Research Articles

Computational Homogenization of Heterogeneous Materials by a Novel Hybrid Numerical Scheme

The Virtual Element Method (VEM) is a recent numerical technique capable of dealing with very general polygonal and polyhedral mesh elements, including irregular or non-convex ones. Because of this feature, the VEM ensures noticeable simplification in the data preparation stage of the analysis, especially for problems whose analysis domain features complex geometries, as in the case of computational micro-mechanics problems. The Boundary Element Method (BEM) is a well known, extensively used and effective numerical technique for the solution of several classes of problems in science and engineering. Due to its underlying formulation, the BEM allows reducing the dimensionality of the problem, resulting in substantial simplification of the pre-processing stage and in the reduction of the computational effort, without jeopardizing the solution accuracy. In this contribution, we explore the possibility of a coupling VEM and BEM for computational homogenization of heterogeneous materials with complex microstructures. The test morphologies consist of unit cells with irregularly shaped inclusions, representative e.g., of a fiber-reinforced polymer composite. The BEM is used to model the inclusions, while the VEM is used to model the surrounding matrix material. Benchmark finite element solutions are used to validate the analysis results.

A comprehensive analysis of auxetic honeycomb sandwich plates with graphene nanoplatelets reinforcement

Nowadays, sandwich plates with cellular core structures are gaining the attention of researchers owing to their outstanding features. For a better insight into auxetic structures, in this study, we propose an excellent computational approach based on polygonal meshes to comprehensively examine the free vibration, buckling and dynamic instability behaviors of the auxetic honeycomb sandwich plate structures. A generalized C0-type higher-order shear deformation theory (C0-HSDT) in conjunction with both Laplace and quadratic serendipity shape functions is employed to approximate the strain fields for polygonal plate elements. The sandwich plate structures are constituted by an auxetic honeycomb core layer with negative Poisson’s ratio and two skin layers reinforced by graphene nanoplatelets (GNPs). Ultra-light features of the plate structures can be obtained by using the auxetic honeycomb cells with negative Poisson’s ratio while the GNPs are embedded into skin layers to enhance the structural stiffness. In order to determine the dynamic instability region of the sandwich plate, Bolotin’s approach is utilized in the current research. Several numerical examples are carried out to investigate the influences of geometrical parameters of auxetic cell and GNPs reinforcement on the structural behaviors. The results obtained in the current research can be considered as benchmark ones to investigate auxetic sandwich plate structures.

Investigation of the excavation of stony soil slopes using the virtual element method

This study improved two parts of the virtual element method (VEM) to ensure that the stability of a stony soil slope, especially under excavation conditions, could be better studied. One of the improvements was the specific realization of the excavation algorithm, including the determination of how to identify polygonal elements that needed to be excavated, and the “deactivate and reactivate polygonal elements” used to calculate the excavation load during the slope excavation process. The second improvement was the calculation of the factor of safety (FOS) of a slope by combining shear strength reduction and the φ−ν inequality so that the stability of the stony soil slope could be better evaluated. The principal goal was to extend the latest developed virtual element method to the analysis and calculation of the excavation of a stony soil slope. Because the VEM was shape independent, it also allowed the number of nodes in an element to be chosen flexibly and fluidly, so the number of nodes could be changed easily during a simulated procedure. Two numerical examples of the one-step excavation of a homogeneous slope and the two-step excavation of a stony soil slope were solved with the improved VEM. The numerical results showed that the VEM could accurately simulate the excavation process of the slope.

Enhanced numerical integration scheme based on image compression techniques: Application to rational polygonal interpolants

Polygonal finite elements offer an increased freedom in terms of mesh generation at the price of more complex, often rational, shape functions. Thus, the numerical integration of rational interpolants over polygonal domains is one of the challenges that needs to be solved. If, additionally, strong discontinuities are present in the integrand, e.g., when employing fictitious domain methods, special integration procedures must be developed. Therefore, we propose to extend the conventional quadtree-decomposition-based integration approach by image compression techniques. In this context, our focus is on unfitted polygonal elements using Wachspress shape functions. In order to assess the performance of the novel integration scheme, we investigate the integration error and the compression rate being related to the reduction in integration points. To this end, the area and the stiffness matrix of a single element are computed using different formulations of the shape functions, i.e., global and local, and partitioning schemes. Finally, the performance of the proposed integration scheme is evaluated by investigating two problems of linear elasticity.

Equal-Order Polygonal Analysis for Fluid Computation in Curved Domain

Our research introduces a novel advanced method for fluid computation in complicated domain. It makes use of an advanced polygonal finite element applying equal-order scheme; within each element. Both velocity and pressure fields are represented by a polygonal basis shape function system. This technique is based on a combination of the equal-order mixed scheme method, the polygon finite element method (PFEM) and the adapted local projection method that is used to stabilize the pressure. In order to simulate the fluid flow, MATLAB is utilized for coding and programming. A numerical example of an incompressible fluid flow governed by Stokes system in curved domain is performed to demonstrate the applicability of our developed polygon element. Besides, through the numerical analysis, we show the high flexibility of PFEM for curved domains.

Simultaneous Smoothing and Untangling of 2D Meshes Based on Explicit Element Geometric Transformation and Element Stitching

Mesh quality can affect both the accuracy and efficiency of numerical solutions. This paper first proposes a geometry-based smoothing and untangling method for 2D meshes based on explicit element geometric transformation and element stitching. A new explicit element geometric transformation (EEGT) operation for polygonal elements is firstly presented. The transformation, if applied iteratively to an arbitrary polygon (even inverted), will improve its regularity and quality. Then a well-designed element stitching scheme is introduced, which is achieved by carefully choosing appropriate element weights to average the temporary nodes obtained by the above individual element transformation. Based on the explicit element geometric transformation and element stitching, a new mesh smoothing and untangling approach for 2D meshes is proposed. The proper choice of averaging weights for element stitching ensures that the elements can be transitioned smoothly and uniformly throughout the calculation domain. Numerical results show that the proposed method is able to produce high-quality meshes with no inverted elements for highly tangled meshes. Besides, the inherent regularity and fine-grained parallelism make it suitable for implementation on Graphic Processor Unit (GPU).

A Genetic Algorithm adaptive homogeneous approach for evaluating settlement-induced cracks in masonry walls

This paper presents a Genetic Algorithm adaptive homogeneous approach aiming at representing the crack patterns induced by ground settlements on masonry walls. This type of damage is a critical issue since it affects all masonry buildings, including those that are not located in seismic-prone areas. The GA-adaptive homogeneous approach here proposed is meant as a tool that overcomes the usual high computational costs requested by the traditional heterogeneous and homogeneous approaches. Here, the considered masonry wall is discretized into a low number of 2D polygonal elements; its displacement field is then determined through a linear programming problem. The actual position of cracks induced by the applied settlement is identified by modifying the initial mesh through an iterative mesh adaptation procedure performed with a Genetic Algorithm (GA); the iterations are carried on until the absolute minimum of the work performed by the reaction forces is attained. In this way, the computational effort needed for identifying the actual crack patterns is dramatically decreased due to the very few unknowns of the problem. The reliability of the GA-adaptive homogeneous approach here proposed is validated against selected benchmarks that come from experimental and numerical results, and is also compared with the traditional heterogeneous and homogeneous approaches. In all the three benchmarks, the GA-adaptive approach offers a satisfying computational efficiency and identifies the actual crack patterns with good accuracy, despite the low number of elements employed in the discretization of the masonry wall. This may pave the way for a broader use of this approach in the analysis of complex masonry structures affected by settlement-induced damages.

On the Six Node Hexagon Elements for Continuum Topology Optimization of Plates Carrying in Plane Loading and Shell Structures Carrying out of Plane Loading

The need of polygonal elements to represent the domain is gaining interest among structural engineers. The objective is to perform static analysis and topology optimization of a given continuum domain using the rational fraction type shape functions of six node hexagonal elements. In this paper, the main focus is to perform the topology optimization of two-dimensional plate structures using Evolutionary Swarm Intelligence Firefly Algorithms (ESIFA) and three-dimensional shell structures using optimality criteria. The optimization of plates carrying in plane loading is performed with minimum weight as objective. Two different types of shell structures are optimized using maximum strain energy as criteria. The optimal distribution of the material in the design domain obtained using six node hexagon elements is compared with the optimal distribution of material obtained using quadrilateral elements. A few problems from the literature have been solved and this study has proved that hexagon element gives better results over traditional quadrilateral elements.

A Posteriori Error Estimates of Virtual Element Method for a Simplified Friction Problem

We study a posteriori error analysis of the virtual element method (VEM) for a simplified friction problem, which is a representative elliptic variational inequality of the second kind. By treating hanging nodes as vertices of polygonal elements, the virtual element method does not require any local mesh post-processing after the adaptive mesh refinement. In this work, residual type error estimators are derived for designing adaptive VEM to solve the simplified friction problem. Furthermore, the reliability and efficiency of the error estimators are proved. Finally, a numerical example is given to verify the theoretical results.

Polygonal topology optimization for Reissner–Mindlin plates

Polygonal finite elements have recently been widely implemented in topology optimization problems due to their great advantages such as highly accurate solutions and flexibility in mesh generation of arbitrary-shaped design domains. In this study, a polygonal topology optimization for Reissner–Mindlin (R–M) plate with minimizing the compliance under a volume constraint is proposed. In order to avoid the critical transverse shear locking phenomenon arising in the R–M plate theory, a simple and efficient locking-free polygonal R–M plate element (PRMn) based on the Timoshenko’s beam assumptions is applied to solve the R–M plate topology optimization problems. In our proposed method, we use the solid isotropic material with penalization model, the standard optimality criteria methods and density filter.

Influence of contact points of helicopter skid landing gears on ground resonance stability

Soft-in-plane rotor systems are susceptible to a self-induced vibration phenomenon called ground resonance. This dynamic instability results from lag motions of the rotor blades coupling with airframe degrees of freedom, while the helicopter is in ground contact. As an addition to slope landing studies in the past and investigations of non-linear landing gear effects, this work focuses on a systematic study of partial skid contact. A ground resonance test environment was created. It encompasses 3D helicopter models with flexible landing gear models using direct finite-element models and applying modal reduction to embed CAD-derived landing gear models. Both approaches are used in a multibody dynamics simulation. The models showed acceptable results for the simulation of non-linear dynamic behaviour including typical non-linear effects like limit cycles. Special focus is given to different methods of contact simulation, using 3D spring–damper elements and polygonal contact elements for multi-directional contacts. The simulations showed two counteracting effects for partial ground contact and time-variant contact conditions. On one hand, the reduction of restoring forces in partial ground contact should lead to more unstable conditions. On the other hand, energy dissipation shows a larger influence on the system stability behaviour after a sudden disturbance. This effect is of high interest for soft, partial landing conditions.

A Fast UTD-Based Method for the Analysis of Multiple Acoustic Diffraction over a Series of Obstacles with Arbitrary Modeling, Height and Spacing

A uniform theory of diffraction (UTD)-based method for analysis of the multiple diffraction of acoustic waves when considering a series of symmetric obstacles with arbitrary modeling, height and spacing is hereby presented. The method, which makes use of graph theory, funicular polygons and Fresnel ellipsoids, proposes a novel approach by which only the relevant obstacles and paths of the scenario under study are considered, therefore simultaneously providing fast and accurate prediction of sound attenuation. The obstacles can be modeled either as knife edges, wedges, wide barriers or cylinders, with some other polygonal diffracting elements, such as doubly inclined, T- or Y-shaped barriers, also considered. In view of the obtained results, this method shows good agreement with previously published formulations and measurements whilst offering better computational efficiency, thus allowing for the consideration of a large number of obstacles.

A polygonal finite element approach for fatigue crack growth analysis of interfacial cracks

The fatigue crack growth analysis of an interfacial cracked structure plays an essential role in the design process as well as the safety assessment for various engineering fields in real life. This paper proposes a novel and effective computational approach based on polygonal meshes for crack growth simulation of interfacial crack problems. Polygonal meshes are fully exploited to provide greater flexibility in mesh generation for complicated domains as well as obtain more accuracy in terms of the numerical results. The twelve crack tip enrichment functions to capture the near-tip oscillating singular field are first implemented for polygonal meshes to improve the obtained results. The Wachspress shape function is applied for arbitrary polygonal elements. The strong discontinuities e.g. crack face, crack tip can be modeled, respectively, via Heaviside and asymptotic crack tip enrichment functions while the material interface discontinuities are defined by signed distance functions. The stress intensity factors (SIFs) of the interfacial crack problems can obtain by using the domain based interaction integral approach while the fatigue life is estimated via the Paris law. The accuracy and efficiency of the present approach are demonstrated through several numerical investigations regarding the crack growth problems in both homogeneous materials and bi-materials. The results which are generated from the proposed approach are compared with other earlier ones as well as the conventional XFEM solutions.

Special polygonal elements for thermal analysis of cellular material containing elliptical holes

A special hybrid polygonal finite element (FE) is constructed for the efficient investigation of two-dimensional heat conduction problem in a cellular material containing a large number of elliptical holes. The special fundamental solution representing the thermal response of a point heat source in an infinite isotropic solid containing a centered inclined elliptical hole is derived for the construction of the special element enclosing the elliptical hole. Thus, the further fine mesh division along the rim of elliptical hole is thoroughly avoided. In the special FE formulation, the special fundamental solution is taken as trial function to construct the intra-element temperature field, which exactly satisfies the governing equation of problem and the insulating condition along the hole boundary, but not the specific temperature and heat flux boundary conditions on the outer boundary of the computing domain and the continuity condition between adjacent elements. To fix this, the temperature field along the element boundary is independently approximated by the conventional FE interpolation function. Combing these two independent fields into the modified double-variable Hellinger–Reissner variational principle leads to a system of stiffness equations including element boundary integrals only, whose computing dimension is reduced by one compared to the domain integrals. More importantly, such integration feature endows the present method great capability of constructing any shaped polygonal element with different numbers of edges and nodes in a unified form with same kernels, and extremely simplifying mesh effort around the elliptical hole. Numerical experiments are performed on special polygonal element meshes for the problems associated with one, two and multiple elliptic holes and the results show a good accuracy and convergence performance for the proposed special element.

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.

Study on heat transfer characteristics of a circular tube with polygonal twisted elements

This paper reported experimental and numerical simulation investigations on heat transfer characteristics of polygonal twisted elements in the constant wall temperature with Reynolds number (Re) ranging from 200 to 1800. In this study, the results revealed that the entransy efficiency (η) increased with the polygonal twisted element’s edge number increasing. Moreover, the highest entransy efficiency η of the quadruple-edge continuous twisted element (QEC) heat exchange tube was 93.97%. Besides, the field synergy performance also showed a positive correlation with the element edge number. A maximum increase of 57% was observed in the field synergy number (Fc) of the QEC insert as compared with the plain tube while the performance evaluation criteria (PEC) had little relationship with the element edge number. In the case of the interlaced twisted elements, both the PEC value and the entransy efficiency η were lower than the corresponding continuous ones. In addition, the Fc values were not apparent, of which the maximum difference was only 7% in the two cases(continuous twisted element and interlaced twisted element). In Re= 1800, when the d/D(the ratio of the twisted element diameter to the inner diameter of the tube) of the QEC heat exchange tube increased, the PEC values and Fc values both increased significantly. The maximum PEC value was 1.62 and the maximum Fc values was 202% higher than the plain tube. Subsequently, the entransy efficiency η got the maximum value in the range of d/D = 0.375-0.75. The maximum entransy efficiency was 91.07%.

Polygonal finite element: A comparison of the stiffness matrix integration methods

This paper aims to determine which numerical integration method shows the best performance when integrating the polygonal finite element stiffness matrix. Hence, numerical comparisons were made between two existing methods, triangulation and quadrangulation, and two new quadrature rules developed by the authors called Polygonal Gauss–Legendre quadrature (PGL) and Rational Polynomial Gauss quadrature (RPG). In this study, PGL11Polygonal Gauss-Legendre quadrature. points and weights were obtained for polynomials of degree 3, 5, 6, 8 and 9, while RPG22Rational Polynomial Gauss quadrature. rules were calculated for rational polynomials of degree 8/8, 10/8 and 16/16. Briefly, the PGL rule showed the best performance, reducing the execution time in 38.6% at best (compared with the triangulation), also, its implementation is much more straightforward.

A polygonal finite element formulation for modeling nearly incompressible materials

The objective of the present paper is to develop a finite element formulation for modeling nearly incompressible materials at large strains using polygonal elements. The present finite element formulation is a simplified version of the three-field mixed formulation and, in particular, it reduces the functional of the internal potential energy by expressing the field of the average volume-change in terms of the displacement field, where the latter is discretized using the Wachspress shape functions. The reduced mixed formulation eliminates the volumetric locking in nearly incompressible materials and enhances the computational efficiency as the static condensation is circumvented. A detailed implementation of the finite element formulation is presented in this study. Also, different example problems, including eigenvalue analysis, nonlinear patch test and other benchmark problems are presented for demonstrating the accuracy and the reliability of the developed formulation for polygonal elements.

A parameterized level set method combined with polygonal finite elements in topology optimization

Level set-based topology optimization methods have received increased attention in the last decade due to the ease with which they can handle topology changes by splitting and merging boundaries. However, most implementations are limited to box-like design domains and cannot easily handle irregular domains that are common in practical engineering applications. In this paper, a modified level set model, which is parameterized and updated with radial basis functions (RBFs), is introduced to solve topology optimization problems in complex design domains. Because RBFs are not necessarily located in a structured manner, the parametrized level set method can be combined effortlessly with unstructured polygonal finite element meshes to easily handle complex design domains and also achieve the high accuracy characteristic of polygonal discretizations. Furthermore, the nucleation capability of the proposed approach along with an approximate reinitialization is shown to obtain more flexible topology changes of the optimal design, and smooth boundaries can be obtained without introducing smoothing scheme. Several numerical examples illustrate the effectiveness of the proposed model.

The polygonal spline thin plate element based on the discrete Kirchhoff theory

As we know, the polygonal elements can do well in simulation of the materials behavior and provide greater flexibility for the meshing of complex geometries. Besides, the hanging nodes can be handled as irregular nodes of polygonal element. Hence, the study on the polygonal element is a very useful and necessary part in the finite element method. However, the research on polygonal plate element is lack. The difficulty of polygonal element is the construction of the interpolation bases.In this paper, by using the spline and B-net method, we construct a polygonal spline thin plate element based on the discrete Kirchhoff theory. The key is to construct a set of spline interpolation bases corresponding to the boundary nodes of the polygonal element including the irregular case. The basic idea is to subdivide the $n$-sided polygon into $n$ subtriangles. We represent the spline functions in each subtriangle as quadratic polynomials in the B-net form. Then the degree of freedoms of the B-net coefficients corresponding to the interior B-net domain points are eliminated by some proper continuous conditions. As a result, we obtain the spline interpolation bases as the shape functions on the polygonal element. Combined with the discrete Kirchhoff theory, the new polygonal thin plate element is constructed and denoted by DKPS element. The nodal degrees of freedom are the displacement and two rotations at each vertex of the polygonal element. It can possess quadratic completeness in the Cartesian coordinates and is valid for both nonconvex and irregular polygonal elements. The mathematical formulas for the element stiffness matrices are also presented, which can be computed directly by the B-net coefficients of the spline bases. Thus there is no need to use the numerical integration in the computation of the element stiffness matrices. Some numerical experiments show that the new element has efficiency and good accuracy even for some badly distorted meshes.