Polygonal Elements Research Articles

The GOGIRA System: An Innovative Method for Landslides Digital Mapping

Landslide mapping techniques have had many improvements in recent decades, the main field of development has been on traditional cartographic techniques and to a lesser extent on indirect numerical cartography. As for Direct Numerical Cartography (DNC), only a few improvements have been made due to the complexity and economic cost of the new technologies. To meet this lack in DNC techniques GOGIRA (Ground Operative-system for GIS Input Remote-data Acquisition), a new system following the GIS (Geographic Information System) scheme, was developed. It is a suite of hardware and software tools, algorithms, and procedures for easier and cheaper DNC. Initial tests conducted on the Quincinetto landslide system (north-western Italy) demonstrated good results in terms of morphometric coherence and precision. A geomorphological map made with GOGIRA was compared with a highly detailed geomorphological map developed with modern tested methods. In conclusion GOGIRA proved to be a valid system for geomorphological DNC when applied to a complex landslide system, considering the early stage of developing results for linear and point mapping was excellent, as for polygonal elements more studies must be conducted to improve accuracy and precision.

Virtual Element Method and Optimal Shape Design in Magnetics

We propose an innovative technique for dealing with optimal shape design problems that exploits the flexibility of the virtual element method (VEM) in handling meshes with general polygonal and polyhedral elements. The shape synthesis of a magnetic pole is considered as the case study.

A new quality preserving polygonal mesh refinement algorithm for Polygonal Element Methods

Mesh adaptivity is a useful tool for efficient solution to partial differential equations in very complex geometries. In the present paper we discuss the use of polygonal mesh refinement in order to tackle two common issues: first, adaptively refine a provided good quality polygonal mesh preserving quality, second, improve the quality of a coarse poor quality polygonal mesh during the refinement process.

A fully discrete plates complex on polygonal meshes with application to the Kirchhoff–Love problem

In this work we develop a novel fully discrete version of the plates complex, an exact Hilbert complex relevant for the mixed formulation of fourth-order problems. The derivation of the discrete complex follows the discrete de Rham paradigm, leading to an arbitrary-order construction that applies to meshes composed of general polygonal elements. The discrete plates complex is then used to derive a novel numerical scheme for Kirchhoff–Love plates, for which a full stability and convergence analysis are performed. Extensive numerical tests complete the exposition.

A general analytical PBEM for solving three-dimensional transient inhomogeneous heat conduction problems with spatially varying heat generation

In the present work, a polygonal boundary element method (PBEM) for solving transient inhomogeneous heat conduction problems with spatially-varying heat generation is developed for the first time. A new general analytical method is proposed and employed in the present PBEM, by which the radial integrals associated with arbitrary spatially-varying density, specific heat, and thermal conductivity can be analytically computed, and the efficiency of the PBEM for solving the transient inhomogeneous problems can be significantly improved. The transient term is dealt with by a finite difference scheme. Three examples are designed to investigate the performance of the proposed method, and the results show that the present method could accurately and efficiently solve transient heat conduction problems with arbitrarily-varying thermal physical properties.

BDDC Preconditioners for Divergence Free Virtual Element Discretizations of the Stokes Equations

The virtual element method (VEM) is a new family of numerical methods for the approximation of partial differential equations, where the geometry of the polytopal mesh elements can be very general. The aim of this article is to extend the balancing domain decomposition by constraints preconditioner to the solution of the saddle-point linear system arising from a VEM discretization of the two-dimensional Stokes equations. Under suitable hypotesis on the choice of the primal unknowns, the preconditioned linear system results symmetric and positive definite, thus the preconditioned conjugate gradient method can be used for its solution. We provide a theoretical convergence analysis estimating the condition number of the preconditioned linear system. Several numerical experiments validate the theoretical estimates, showing the scalability and quasi-optimality of the method proposed. Moreover, the solver exhibits a robust behavior with respect to the shape of the polygonal mesh elements. We also show that a faster convergence could be achieved with an easy to implement coarse space, slightly larger than the minimal one covered by the theory.

Slope stability analysis based on SBFEM and multistage polytree-based refinement algorithms

A fresh slope stability analysis method is proposed in this paper, with combining scaled boundary finite element method (SBFEM), strength reduction method (SRM) and polytree-based mesh refinement algorithm. Firstly, SBFEM and SRM (SBFEM-SRM) formulation are innovatively combined to conduct the slope stability analysis with a coarse analysis mesh. Secondly, a shear-strain-based refinement indicator is introduced to determine the elements that need to be refined, then multistage local mesh refinement can be implemented recursively on-demand via the independently developed polytree-based refinement algorithm. The accuracy and superiority of SBFEM-SRM are firstly verified using a simple slope example. The effectiveness and applicability of SBFEM-SRM combined with the polytree-based refinement algorithm is demonstrated by using two examples, including a multilayered nonhomogeneous soli slope and an actual test embankment. The advantages of different methods can be inherited in the presented method, wherein flexible polygonal elements can be solved directly, and making a strong adaptability for complex geometry equipped. Moreover, the computational burden would be alleviated efficiently without compromising accuracy, benefiting from local adaptive mesh refinement strategy. And inspired by this, the proposed method can be briefly extended to 3D slope stability analysis, wherein the application performance will be more significant.

A free‐cutting mesh strategy for optimal shape synthesis in magnetics

The authors propose an innovative technique for dealing with optimal shape design problems that exploits the flexibility of the virtual element method in generating meshes composed of general polygonal and polyhedral elements. Virtual element method and finite element method can coexist on the same discretized domain; therefore, the possibility of dealing with hanging nodes and gluing sub-domain meshes is ensured. Accordingly, the shape synthesis of a magnetic pole is considered as the case study. It is shown that the proposed technique is effective in handling the shape variations dictated by an algorithm of evolutionary optimisation.

A Conforming Quadratic Polygonal Element and Its Application to Stokes Equations

A Conforming Quadratic Polygonal Element and Its Application to Stokes Equations

Virtual element method for modeling the deformation of multiphase composites

In this paper, we study applications of the virtual element method (VEM) for simulating the deformation of multiphase composites. The VEM is a Galerkin approach that is applicable to meshes that consist of arbitrarily-shaped polygonal and polyhedral (simple and nonsimple) elements. In the VEM, the basis functions are defined as the solution of a local elliptic partial differential equation, and are never explicitly computed in the implementation of the method. The stiffness matrix of each element is built by using the elliptic projection operator of the internal virtual work (bilinear form) and it consists of two terms: a consistency term that is exactly computed (linear patch test is satisfied) and a correction term (ensures stability) that is orthogonal to affine displacement fields and has the right scaling. The VEM simplifies mesh generation for a multiphase composite: a stiff inclusion can be modeled using a single polygonal or polyhedral element. Attributes of the virtual element approach are highlighted through comparisons with Voronoi-cell lattice models, which provide discrete representations of material structure. The comparisons involve a suite of two-dimensional linear elastic problems: patch test, axisymmetric circular inclusion problem, and the deformation of a three-phase composite. The simulations demonstrate the accuracy and flexibility of the virtual element method.

Very high-order accurate finite volume scheme for the steady-state incompressible Navier–Stokes equations with polygonal meshes on arbitrary curved boundaries

The numerical solution of the incompressible Navier–Stokes equations raises challenging numerical issues on the development of accurate and robust discretisation techniques due to the div-grad duality. Significant advances have been achieved in the context of high-order of convergence methods, but many questions remain unsolved in the quest of practical, accurate, and efficient approaches. In particular, the discretisation of problems with arbitrary curved boundaries has been the subject of intensive research, where the conventional practice consists in employing curved meshes to avoid accuracy deterioration. The present work proposes a novel approach, the reconstruction for off-site data method, which employs polygonal meshes to approximate arbitrary smooth curved boundaries with linear piecewise elements. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed on the polygonal mesh elements. Boundary conditions are taken into account via polynomial reconstructions with specific linear constraints defined for a set of points on the physical boundary. The very-high order of convergence is preserved without relying on the full curve parametrisation, avoiding the limitations of curved mesh approaches, such as sophisticated mesh generation algorithms, cumbersome quadrature rules, and complex non-linear transformations. A comprehensive verification benchmark is provided, with numerical test cases for several fluid flow problems, to demonstrate the capability of the proposed approach to achieve very high-orders of convergence.

Dynamic Fracture Modeling of Impact Test Specimens by the Polygon Scaled Boundary Finite Element Method

In the polygon scaled boundary finite element method, the modeling of crack can be simplified and the stress intensity factors are extracted from the semi-analytical solution directly. These salient features are applied to develop an efficient method to model the impact test specimens, one of which occurs without crack propagation, the other with crack propagation. The notched bend specimens are discretized by polygon elements with no local refinement around the crack tip. The mass and stiffness matrix of polygon elements are derived based on the SBFEM, and then the elastodynamic equations of the global system are established. The Newmark integration method is applied to obtain the dynamic responses of the specimens. For the case with crack propagation, an efficient local remeshing algorithm is used to track the crack tip and model the crack propagation. The dynamic stress intensity factors are computed directly from the instantaneous displacement field and crack velocity. The numerical results of the two specimens correspond well with the experimental data and other numerical results reported in the literature. The effects of the time step, mesh density and damping coefficient are also investigated. Moreover, the displacement and stress contours are extracted from the dynamic solution, which is helpful for the interpretation of the experimental observations.

A Novel Solution for Seepage Problems Implemented in the Abaqus UEL Based on the Polygonal Scaled Boundary Finite Element Method

The scaled boundary finite element method (SBFEM) is a semianalytical computational scheme based on the characteristics of the finite element method (FEM) and boundary element method that combines their respective advantages. In this paper, the SBFEM and polygonal mesh technique are integrated into a new approach to solve steady-state and transient seepage problems. The proposed method is implemented in Abaqus employing a user-defined element (UEL). A detailed implementation of the procedure is presented in which the UEL element is defined, the internal variables RHS and AMATRX are updated, and the stiffness/mass matrix is solved using eigenvalue decomposition. Several benchmark problems are solved to verify the proposed implementation. The results show that the polygonal element of the polygonal SBFEM (PSBFEM) is more accurate than the standard FEM element of the same element size. For transient problems, the results for the PSBFEM and FEM are in excellent agreement. Hence, the proposed method is robust and accurate for solving steady-state and transient seepage problems. The developed UEL source code and the associated input files can be downloaded from GitHub.

An n-sided polygonal selective smoothed finite element method for nearly incompressible visco-hyperelastic soft materials

In this article, an n-sided polygonal smoothed finite element (nSFEM) is formulated for dynamic analyses of nonlinear problems of visco-hyperelastic materials undergoing large deformations. Two types of smoothing domains are constructed for mesh of n-sided polygonal elements. Using the gradient smoothing technique, the calculation of the strain-displacement matrix requires only the shape function values rather than the derivatives of shape function. The simple averaging point interpolation technique is used to calculate the values of the shape functions on the boundary of different smoothing domains. The Total Lagrangian formulation is used to deal with the nonlinear dynamic analysis of soft materials with finite strain. The polygonal FEM with Wachspress coordinate is also implemented for comparison. To overcome volumetric locking, selective smoothing domain scheme is used to calculate the smoothing nodal internal force. To analyze the time-dependent mechanical behavior of soft materials, a constitutive update algorithm for explicit time-integration is implemented based on the generalized Maxwell visco-hyperelastic model. A number of numerical examples are presented to demonstrate the high precision and super convergence, and robustness of the present nS-FEM for incompressible visco-hyperelastic problems.

Expansion of olive orchards and their impact on the cultivation and landscape through a case study in the countryside of Cordoba (Spain)

The sustainability of farming systems has been enhanced by legislation on different scales, but at the same time these policies also promote more productive systems through farming intensification (e.g., use of irrigation or high tree densities). This is the case of olive orchard expansion on cereal cropland in recent decades. This study analyses the impact of this expansion on orchard characteristics and landscape elements in a case study in the ’campiña‘ of Cordoba in Southern Spain based on the evolution of their surface and typologies during the period from 2005 to 2018. Our results show that olive orchards doubled their surface after the 13-year period, from 7997.8 to 16,447.6 ha. On average the new orchards tended to have higher plant density and a more frequent use of irrigation in the study period. Despite this trend towards intensification, the current situation shows a majority of rainfed (76.4%) and medium tree densities, 120–200 trees/ha, (42.7%) of the area. Nevertheless, newly intensified orchards are arising in the region, resulting in a mosaic of orchards of different characteristics (slope, tree density, soil type) and agricultural managements (irrigation, ground cover vegetation).In addition, this characterization was complemented with an inventory of the existing semi-natural elements associated with these orchards to identify the current state of the regional agricultural landscape. A total number of 507 isolated trees and different linear and polygonal landscape elements (343.9 km and 714.0 ha, respectively), mainly segmented, were inventoried. From these polygonal landscape elements, a significant fraction (e.g., slopes, gullies, water banks and non-productive strips/faces) remains unvegetated (57%). Therefore, these elements must be considered in multiscale agricultural policies as potential restoration areas to enhance ecosystem service provisioning.

Quadratic serendipity element shape functions on general planar polygons

This paper proposes a method for the construction of quadratic serendipity element (QSE) shape functions on planar convex and concave polygons. Existing approaches for constructing QSE shape functions are linear combinations of the pair-wise products of generalized barycentric coordinates with linear precision, restricted to the convex polygonal domain or resort to numerical optimization. We extend the construction to general polygons with no more than three collinear consecutive vertices. This is done by defining coefficients of the linear combination as the oriented area of triangles with vertices from the polygonal domain, which can be either convex or concave. The proposed shape functions possess linear to quadratic precision. We prove the interpolation error estimates for mean value coordinate-based QSE shape functions on convex and concave polygonal domains satisfying a set of geometric constraints for standard finite element analysis. We also tailor a polygonal mesh generation scheme that improves the uniformity and avoids short edges of Voronoi diagrams for their use in the QSE-based polygonal finite element computation. Numerical tests for the 2D Poisson equations on various domains are presented, demonstrating the optimal convergence rates in both the L2-norm and the H1-seminorm.

Polygonal composite elements for stress-constrained topology optimization of nearly incompressible materials

We in this paper propose an efficient and vigorous approach based on flexible polygonal meshes for solving stress-constrained topology optimization problems involving both compressible and nearly incompressible materials. The core idea is to employ a polygonal composite finite element technique to deal with volumetric locking phenomena in nearly incompressible material limit. Then, to effectively solve topology optimization problems with local stress constraints, an augmented Lagrangian technique that allows treating problems with a large number of constraints is exploited. By employing the augmented Lagrangian method, the solution of the original optimization problem with a lot of constraints could be obtained by solving a series of unconstrained optimization ones and updating the design variables based on the method of moving asymptotes. The significant contribution of this work is to promote a unified formulation for stress-constrained topology optimization possessing the following salient features: (a) being valid for both compressible and nearly incompressible materials; (b) being valid for triangular, quadrilateral and polygonal elements; (c) sole displacement field involved; and (d) no constraint aggregation technique required. Through several numerical examples, for the first time, distinguishing and intriguing features of the optimized topology for nearly incompressible materials with stress constraints are exhibited in the present work.

Thermoelastic fracture analysis of functionally graded materials using the scaled boundary finite element method

The scaled boundary finite element method is extended to model fracture in functionally graded materials (FGM) under coupled thermo-mechanical loads. The governing equations of coupled thermo-mechanical equilibrium are discretized using scaled boundary shape functions enriched with the thermal load terms. The material gradient is modeled as a series of power functions, and the stiffness matrix is calculated semi-analytically. Stress intensity factors and T−stress are directly calculated from their definition without any need for additional post-processing techniques. Arbitrary-sided polygon elements are employed for flexible mesh generation. Several numerical examples for isotropic and orthotropic FGMs are presented to validate the proposed technique.

Robust and scalable adaptive BDDC preconditioners for virtual element discretizations of elliptic partial differential equations in mixed form

The Virtual Element Method (VEM) is a recent numerical technology for the solution of partial differential equations on computational grids constituted by polygonal or polyhedral elements of very general shape. The aim of this work is to develop effective linear solvers for a general order VEM approximation designed to approximate three-dimensional scalar elliptic equations in mixed form. The proposed Balancing Domain Decomposition by Constraints (BDDC) preconditioner allows to use conjugate gradient iterations, albeit the algebraic linear systems arising from the discretization of the problem are indefinite, ill-conditioned, and of saddle point nature. The condition number of the resulting positive definite preconditioned system is adaptively controlled by means of deluxe scaling operators and suitable local generalized eigenvalue problems for the selection of optimal primal constraints. Numerical results confirm the theoretical estimates and the reliability of the adaptive procedure, with the experimental condition numbers always very close to the prescribed adaptive tolerance parameter. The scalability and quasi-optimality of the preconditioner are demonstrated, and the performances of the proposed solver are compared with state-of-the-art parallel direct solvers and block preconditioning techniques in a distributed memory setting.

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.