Polygonal Elements Research Articles

Polygonal plate element method for free vibration analysis using an efficient alpha (α)-assumed rotations and shear strains

This paper investigates the performance of a recently proposed polygonal plate element with alpha (α)-assumed rotations and shear strains, referred to as αARS-Poly, in free vibration analysis. The αARS-Poly element utilizes a simple and efficient approach involving a scaling factor (α) to enhance the accuracy of assumed rotations and shear strains. To fully explore the advantages of this element, we undertake a comprehensive analysis of free vibration in plate structures using a range of models with complex geometries. Numerical results demonstrate that the αARS-Poly element offers stability and reliability within smooth mode shapes. Furthermore, it significantly outperforms the previous polygonal Reissner-Mindlin plate element with piecewise-linear shape functions (PRMn-PL), achieving frequencies that closely match reference solutions, thereby validating its accuracy and robustness for dynamic applications.

Adaptive scaled boundary finite element method for hydrogen assisted cracking with phase field model

This study introduces a numerical framework for analysing hydrogen assisted cracking, employing the scaled boundary finite element method. This is the first instance where the scaled boundary finite element method is employed to model hydrogen embrittlement. The phase field model is utilized to simulate defects, complemented by adaptive meshing using polytree mesh. The ability of the scaled boundary finite element method to treat polygonal elements assists in mitigating the problem of hanging nodes that arises from polytree decomposition. The adaptive framework is developed to predict crack propagation using an initial unstructured quadrilateral mesh generated from any commercial software. The hydrogen atom concentration depends on the hydrostatic stress gradient, which is calculated by interpolating nodal hydrostatic stress with scaled boundary shape functions and taking the gradient. A staggered solution approach is adopted to concurrently tackle hydrogen transport, elasticity, and phase field equations. The methodology is validated using Mode-I edge notch specimens and analysing crack propagation from corrosion pits, with results demonstrating close agreement with existing data. Subsequently, the framework is extended to address more intricate scenarios, such as crack propagation in X-shaped plates and hydrogen transmission in tanks. As the last example, more advanced image based fracture analysis of weld structure is investigated. This example studies the possibility of analysing the hydrogen diffusion directly from TEM imagery. Moreover, we investigate how hydrogen concentration influences structural failure.

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.

Numerical integration in the virtual element method with the scaled boundary cubature scheme

AbstractThe virtual element method (VEM) is a stabilized Galerkin method on meshes that consist of arbitrary (convex and nonconvex) polygonal and polyhedral elements. A crucial ingredient in the implementation of low‐ and high‐order VEM is the numerical integration of monomials and nonpolynomial functions over such elements. In this article, we apply the recently proposed scaled boundary cubature (SBC) scheme to compute the weak form integrals in various virtual element formulations over polygonal and polyhedral meshes. In doing so, we demonstrate the flexibility of the approach and the accuracy that it delivers on a broad suite of boundary‐value problems in 2D and 3D over polytopes with affine faces as well as on elements with curved boundaries. In addition, the use of the SBC scheme is exemplified in an enriched Poisson formulation of the VEM in which weakly singular functions are required to be integrated. This study establishes the SBC method as a simple, accurate and efficient integration scheme for use in the VEM.

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).

Thermal analysis of a perforated laminated plate with multiple random elliptical holes by specially formulated finite elements

The high-efficient and accurate analysis of anisotropic composite media weakened by a large amount of holes or cuts is still a great challenge in practical composite engineering. In this work, specially-proposed polygonal finite element enclosing an arbitrarily oriented elliptical hole in general anisotropic composite media is formulated for thermal analysis in the framework of hybrid finite element theory. Two independent temperature field approximations are defined for the present special element: Intra-element temperature field and element frame temperature field. The former defined inside the element is written in the form of truncated summation of the weighted special fundamental solutions, which are derived using the Stroh complex theory and conformal mapping whilst the latter is implemented by the conventional shape function interpolation. Then the hybrid variational functional is employed to combine them together to generate an element stiffness equation with respect to nodal temperature and an optional relationship between the interpolation coefficients. The element stiffness matrix is symmetric and involves element boundary integrals only. Compared to the traditional finite element, the distinctive element-boundary-integral and enclosed-elliptical-hole features make the present element significantly improve the meshing effort around an elliptical hole by reducing the number of elements and decreasing the meshing difficulty, and more importantly, effectively capture the critical variation of physical fields around the elliptical hole without locally-refined meshes. Moreover, the present special element can be incorporated directly into the conventional finite element system without any difficulty for extensive analysis. Finally, several typical examples are simulated by the present special elements to demonstrate their effectiveness.

A hierarchical shape description approach and its application in similarity measurement of polygon entities

Spatial similarity provides an important basis for geographic information processing and is widely applied in multi-source data fusion and update, data retrieval and query, and cartographic generalization. To address the shape description and similarity measurement of polygon entities, this study presents a new hierarchical shape description approach and examines its application in similarity measurement of polygon entities. Using the rotation and segmentation methods, we first constructed a hierarchical shape description model for target polygon entities, followed by measurement of global and hierarchical shape description of polygon entities, respectively, using the farthest-point-distance and geometric feature description methods. Finally, we constructed a comprehensive similarity measurement model through a weighted integration of position, size, direction, and shape. The hierarchical shape description approach proposed in this paper can be applied to the shape similarity measurement of polygon elements, similarity measurement after spatial object simplification, and multi-scale polygon entity matching. The experimental results showed that the hierarchical shape description approach and similarity measurement model are able to effectively measure spatial similarity between different polygon entities, and have obtained good application results.

Crack Propagation in Heterogeneous Gravity Dams Due to Overflow Using Polygonal Grain-Based Distinct Element Method

Crack Propagation in Heterogeneous Gravity Dams Due to Overflow Using Polygonal Grain-Based Distinct Element Method

A new strain-based pentagonal membrane finite element for solid mechanics problems

Polygonal finite elements remain an attractive option in finite element analysis due to their flexibility in modeling arbitrary shapes compared to triangles. In this study, a pentagonal membrane element was developed with the strain approach for the first time. The element possesses invariance, and the equilibrium constraint was applied to the assumed strain field using corrective coefficients. Inspired by the advancing front technique, a pentagonal mesh was generated, and the mesh quality was enhanced with Laplacian smoothing. The performance of the developed pentagonal element was assessed in a few numerical tests, and the results revealed its suitability in modeling the bending of beams. Besides, the numerical results are enhanced when pentagonal elements are used in mesh transitions along boundaries to smoothen curved edges and capture distributed loads.

Um método híbrido de otimização topológica aplicado em estruturas de concreto armado utilizando elementos finitos poligonais

Abstract This work introduces a new alternative to obtain strut-and-tie models using the hybrid topology optimization method, which is already proposed in the technical literature and is refined here to use polygonal finite elements and accelerate the solution of the material nonlinearity problem. In this method, concrete is approached as a continuum, using polygonal two-dimensional finite elements, and steel bars as truss elements, using one-dimensional finite elements with two nodes. For a closer representation of reality, an orthotropic constitutive model for concrete was implemented considering different compression and tensile stiffness values, which is one of the advantages of the model. Further, the hybrid method limits the final layout of steel bars, thereby generating better structures from a constructive point of view, while allowing greater freedom for the shape and concrete strut slope. However, this method is more complex, and it increases the computational cost, which was substantially minimized through the implementation of an algorithm. Results obtained for some domains were very close to the results of other methodologies; however, small differences were noted that may be relevant to the final result. Other domains showed results with greater differences, thereby significantly changing the final strut-and-tie model and presenting a new structural design alternative.

A polygonal topology optimization method based on the alternating active-phase algorithm

<abstract> <p>We propose a polygonal topology optimization method combined with the alternating active-phase algorithm to address the multi-material problems. During the process of topology optimization, the polygonal elements generated by signed distance functions are utilized to discretize the structural design domain. The volume fraction of each material is considered as a design variable and mapped to its corresponding element variable through a filtering matrix. This method is used to solve a multi-material structural topology optimization problem of minimizing compliance, in which a descriptive model is established by using the alternating active-phase algorithm and the solid isotropic microstructure with penalty theory. This method can accomplish the topology optimization of multi-material structures with complex curve boundaries, eliminate the phenomena of checkerboard patterns and a one-node connection, and avoid sensitivity filtering. In addition, this method possesses fine numerical stability and high calculation accuracy compared to the topology optimization methods that use quadrilateral elements or triangle elements. The effectiveness and feasibility of this method are demonstrated through several commonly used and representative numerical examples.</p> </abstract>

Analysis of free vibration in thin-walled plates using an enhanced polygonal plate element with selective interpolation approach

A highly efficient selective element-interior interpolation approach is proposed that impressively enhances the performance of a spatially isotropic polygonal element (SI-ARS-Poly) and provides outstanding solutions in the displacement and energy norms for the static analysis of thin-walled plate structures (Nguyen and Phan (2023) [22]). In this approach, the element-edge rotations and shear strains are respectively interpolated into interior of polygonal element using linear and quadratic approximations. In order to comprehensively explore the advantages of this element, we extend it to free vibration analysis. Numerical results demonstrate that the proposed approach is stable in the free vibration analysis of plates with smoothed mode shapes. Notably, its solutions for free vibration remain robust in the face of mesh distortion. In comparison with other existing polygonal elements, it is superior with the lowest frequencies closely approximate to the exact/reference ones.

N-sided polygonal cell-based smoothed finite element method (nCS-FEM) based on Wachspress shape function for modal analysis

In this article, an n-sided polygonal cell-based smoothed finite element (nCS-FEM) based on the Wachspress shape function is formulated for free and forced vibration of solid structures. Cell-based smoothing domains are constructed in n-sided polygonal element mesh. Using the gradient smoothing technique, the computation of the strain-displacement matrix requires only the value of the Wachspress shape function, not the derivatives of the shape function or the mapping relationship. The Lanczos algorithm is used to solve eigenvalue problems that produces vibration modes of a given structure, and the vibration modes and modal superposition techniques are used to obtain transient dynamic responses of structures subjected to arbitrary dynamics forces. The nCS-FEM is applied to solve a number of solid structures for modal analysis and compared with the Polygonal FEM, nES-FEM and ABAQUS, it is found that nCS-FEM has high precision, high efficiency and super convergence.

A monotone finite volume element scheme for diffusion equations on arbitrary polygonal grids

We develop a monotone finite volume element scheme for the diffusion problem on arbitrary polygonal grids inspired by the scheme in the literature ([35]). The main contributions of this paper include three aspects. Firstly, some auxiliary lines or points are introduced for any polygonal element so that it can be partitioned into two or more triangles, and one novel type of its control volume is concomitantly designed. In this paper, we mainly discuss quadrilateral grids. Secondly, some piecewise linear finite element spaces are imported for the trial spaces. Fictitious elements, continuous extensions and the nonlinear two-point flux idea can all be inherited from the triangular case. Therefore, the monotonicity of the scheme is easily proved only by the assembling of some triangles' element stiff matrices. Thirdly, the agreeable adaptability for distorted grids is completely borned. The strict convexity restriction on the grids can also be removed, the concave or severely distorted polygonal (such as quadrilateral) grids are transformed as the convex and general regular triangles. Finally, numerical results confirm its monotonicity, accuracy and stability.

An adaptive pseudo-lower bound limit analysis for fracture structures

In this study, we present a new approach to determine the limit loads of fracture structures by using the pseudo-lower bound method with adaptive quadtree meshes. A significant advantage of the pseudo-lower bound method is that calculations done with the stress field immediately solve the volumetric locking problem. Additionally, by defining quadtree meshes in the context of polygonal elements, it is possible to handle the technique of adaptive quadtree meshes, which poses a significant challenge in traditional finite elements due to the presence of hanging nodes during the refinement procedure. The von Mises yield criterion is employed, and structures containing cracks are assumed to have plane stress or plane strain conditions. The static form of the limit analysis is changed into second-order cone programming (SOCP), and the Mosek tool is then used to solve it. Numerical validation is used to illustrate the reliability and effectiveness of the proposed approach.

Boundary Element Method for 3D Laplace and Stokes Flow Problems with Analytical Technique

The analytical technique of element integral in the boundary element method for 3D Laplace and Stokes flow problems is investigated. The boundary integral equations are discretized with constant elements. The local coordinate transformation and polar coordinate transformation techniques are adopted to induce the analytical expressions of element integrals. For 3D Laplace problems, the analytical integral formulae, which are valid for an arbitrary source point, are derived. For 3D Stokes flow problems, analytical evaluation of the integrals, in which the source point and the boundary element are in the same plane, are proposed. These analytical formulae are applicable to arbitrary convex polygonal planar elements. The purpose of numerical examples is to make the case that the analytical expressions are accurate.

Numerical investigation of the influence of mineral mesostructure on quasi-static compressive behaviors of granite using a breakable grain-based model

The mesostructure of brittle rocks, such as granite, plays a vital role in determining their mechanical properties and failure mode. Understanding the influence of rock mesostructure on mechanical behavior requires a realistic representation of grain size distribution, grain shape, and average grain size. In this study, we developed a breakable polygonal discrete element model that incorporates mineralogical composition, grain size distributions, and grain shape to simulate the rock mesostructure. Numerical specimens with varying mesostructures were created to represent different grain size, shape, and distribution characteristics. Quasi-static uniaxial compressive loading tests were conducted on these specimens to analyze their peak strength and macroscopic failure modes. The results revealed a strong linear relationship between the quasi-static compressive strength of the rock and mesostructure parameters, including average grain size, grain size coefficient, and grain roundness. Additionally, the simulation results demonstrated that the rock mesostructure significantly influenced the quasi-static compression failure mode. The proposed breakable polygonal discrete element model has the potential to predict the macroscopic behavior of brittle rocks accurately. It provides a reliable method for studying the effect of mesostructure on the quasi-static compressive mechanical behavior of rocks.

On the stability of mixed polygonal finite element formulations in nonlinear analysis

AbstractThis article discusses the accuracy and stability of the pressure field in nonlinear mixed displacement‐pressure finite element formulations in solid mechanics. We focus on two‐dimensional mixed polygonal finite element formulations with linear displacement and constant pressure approximations in particular. The inf‐sup stability of these formulations is assessed and compared with classical mixed finite element formulations. An analytical proof is presented, which concludes that the occurrence of spurious pressure modes depends on the chosen meshing strategy. It is shown that these spurious modes are successfully suppressed on any Voronoi mesh in both linear elasticity and nonlinear hyperelasticity without the need for any kind of stabilization. Several linear and nonlinear nearly‐incompressible examples with different discretization strategies and boundary conditions are considered to validate the analytical proof. A mixed polygonal finite element formulation based on the scaled boundary parameterization is used to approximate the field variables, however, the derivations presented herein hold for any lowest‐order mixed polygonal finite element formulation. The nonexistence of checkerboard modes on linear elastic Voronoi discretizations is shown graphically. By evaluating the incremental pressure in each Newton–Raphson iteration, the stabilization effect of the Voronoi discretization is demonstrated for nonlinear problems. In addition, the analytical proof is validated by the numerical (generalized) inf‐sup test.

A polygonal finite element method for shakedown analysis of structures

A polygonal finite element method for shakedown analysis of structures

Shear locking free polygonal elements for the analysis of functionally graded plates using (n + 1) integration scheme and Reissner-Mindlin theory

A new shear-locking free polygonal elements is proposed for static bending and free vibration analysis of thin/thick functionally graded material plates. The domain is discretized with arbitrary polygons and Wachspress interpolants are employed for approximating the unknowns. The plate kinematics is based on Reissner-Mindlin plate theory and shear locking syndrome is suppressed by employing the novel n + 1 integration scheme based on Richardson extrapolation technique. Within this, a two stage sampling technique is introduced with the Weighted Richardson scheme to compute the system matrices. From the systematic study, it is inferred that the proposed scheme passes patch tests and yields accurate results.