Arbitrary Polygonal Meshes Research Articles

Virtual element discretization method to optimal control problem governed by Stokes equations with pointwise control constraint on arbitrary polygonal meshes

In this paper we investigate virtual element discretization of optimal control problem governed by Stokes equations. Based on the strategy of first-discretize-then-optimize we build up the virtual element discrete scheme and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L2 and H1 norms are derived. Numerical experiments are presented to verify the theoretical findings.

A Discontinuous Galerkin Finite Element Method with Physical Modal Basis for the Neutron Transport Equation on Arbitrary Polygonal Meshes

The deterministic methods are efficient for solving the neutron transport equation (NTE), but suffer from discretization errors. The increasingly complex geometric models make spatial discretization errors the primary source of discretization errors. Considering that spatial discretization errors come from inaccurate geometric shape descriptions and low-accuracy numerical schemes, this paper develops a Discontinuous Galerkin Finite Element Method for the NTE on unstructured polygonal meshes to reduce spatial discretization errors. In this method, the physical modal basis is adopted to handle the polygonal mesh and to achieve high-order accuracy in a uniform and efficient way. The numerical results of various fixed-source and k-eigenvalue benchmarks demonstrate that the method developed in this paper can give accurate solutions on polygonal meshes with high convergence rates.

The virtual element method for solving two-dimensional fractional cable equation on general polygonal meshes

In this paper, the conforming virtual element method (VEM) is considered to solve the two-dimensional fractional cable equation involving two Riemann–Liouville fractional derivatives. We adopt the Backward Euler Method and the classical scheme for the numerical discrete scheme of the time derivative. Meanwhile, the conforming VEM, which is generated for arbitrary order of accuracy and the arbitrary polygonal meshes, is analysed for the discretization of the spatial direction. Based on the energy projection operator, the fully discrete formula is proved to be unconditionally stable, and the optimal convergence results are derived with regard to the -norm in detail. Finally, some numerical experiments are implemented to verify the theoretical results.

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

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

Positivity‐preserving scheme without separated method for 2D convection–diffusion equations on polygonal meshes

AbstractWe present a finite volume scheme for 2D steady convection–diffusion equations on arbitrary convex polygonal meshes. The method uses auxiliary unknowns on the cell‐edge and cell‐node of the mesh, and the continuous flux is computed to be a two‐point nonlinear flux. Since the transpose of the coefficient matrix is an M‐matrix, it is guaranteed that the scheme is positive. We propose a new strategy, which aims that providing a scheme without separated method to compute the value on the cell‐edge without previous reconstruct. Numerical results show that our scheme is positive on polygonal meshes for both diffusion‐dominated and convection‐dominated problems and has second‐order accuracy for solutions.

A general construction of spherical barycentric coordinates and applications

We provide a general method for defining and efficiently computing barycentric coordinates with respect to polygons on the unit sphere. More precisely, we develop a novel explicit construction which allows us to compute the spherical barycentric coordinates from their 2D-Euclidean counterparts. In particular, we give two interesting families of spherical coordinates, one is defined for convex and non-convex spherical polygons. An interesting consequence is the possibility to construct new 3D barycentric coordinates for arbitrary polygonal meshes. Furthermore, we present an alternative construction for spherical barycentric coordinates with help of 3D barycentric coordinates for closed triangular meshes. This construction is extended to arbitrary dimensions. We show that our spherical and 3D coordinates are widely applicable to many domains. We give several examples related to spherical blending, space deformations and shape morphing in 3D.

A mixed polygonal finite element formulation for nearly-incompressible finite elasticity

In this work, we present a mixed formulation based on the scaled boundary parameterization for the analysis of nearly-incompressible hyperelastic materials at finite strains. The formulation leads to a polygonal element with an arbitrary number of edges. The element domain is parameterized in both the boundary and scaling directions and necessitates the definition of a scaling center. Interpolation functions are introduced in both directions, which allow for an application to physically and geometrically nonlinear problems. This is an extension to the original scaled boundary finite element method (SBFEM), in which an analytical solution in the scaling direction is employed. Whereas the SBFEM approach is restricted to star-convex element geometries, the present approach allows for non-star-convex polygonal element shapes. A mixed displacement-pressure formulation is proposed to alleviate volumetric locking, which typically occurs in case of nearly-incompressible material behavior. Several numerical examples demonstrate that the proposed approach is applicable to arbitrary polygonal meshes, such as Voronoi meshes. The results demonstrate an excellent behavior in terms of computational efficiency compared to standard quadrilateral elements and to the virtual element method (VEM). The ability of the present formulation to alleviate volumetric locking is shown and it is demonstrated that a higher accuracy can be reached on Voronoi meshes compared to quadrilateral discretizations.

Alpha (α) assumed rotations and shear strains for spatially isotropic polygonal Reissner-Mindlin plate elements (αARS-Poly)

This paper proposes simple and efficient alpha assumed rotations and shear strains for polygonal plate elements, named αARS-Poly. In the αARS approach, an alternative assumption of the tangent rotations along element boundaries is applied by using the approximation of the rotations in Timoshenko's beam theory. Then, the quadratic term of this assumed field is linearly scaled up by adding an artificial positive scaling factor α>0. Through examination of the relative errors in the energy (s-) norm regarding α in numerical experiences, the value α=0.5 can be chosen as a general-fixed value that possibly achieves the optimal relative errors of the s-norm. The value α=0.5 seems to be not only mesh-independent but also problem-independent. The αARS-Poly element using α=0.5 passes all critical tests (spatially isotropic, zero-energy mode, and bending path tests) for a finite plate element which ensures the element orientation-independent property, solution stability, and free shear-locking in the thin plate limit. The implementation of the αARS-Poly element is straightforward through a unified form of the stiffness matrix for all arbitrary convex-shaped polygonal meshes. Numerical results show that the proposed element achieves high reliable and optimal results with uniform and excellent convergent rates in the static and free vibration analyses.

Htex

We introduce per-halfedge texturing (Htex) a GPU-friendly method for texturing arbitrary polygon-meshes without an explicit parameterization. Htex builds upon the insight that halfedges encode an intrinsic triangulation for polygon meshes, where each halfedge spans a unique triangle with direct adjacency information. Rather than storing a separate texture per face of the input mesh as is done by previous parameterization-free texturing methods, Htex stores a square texture for each halfedge and its twin. We show that this simple change from face to halfedge induces two important properties for high performance parameterization-free texturing. First, Htex natively supports arbitrary polygons without requiring dedicated code for, e.g, non-quad faces. Second, Htex leads to a straightforward and efficient GPU implementation that uses only three texture-fetches per halfedge to produce continuous texturing across the entire mesh. We demonstrate the effectiveness of Htex by rendering production assets in real time.

Dynamic crack face contact and propagation simulation based on the scaled boundary finite element method

This paper extends the static contact problem of crack faces to include dynamic contact and crack propagation based on the relevant literature (Zhang et al., 2018). The scaled boundary finite element method is used to solve dynamic crack face contact and crack propagation problems. Based on the scaled boundary finite element shape function considering side-face loads, the system dynamic equilibrium equation and related characteristic matrix are presented. The Lagrange multiplier method is used to establish the contact model between the crack faces. The meshing and remeshing algorithm is carried out by using arbitrary polygonal meshes. When a crack propagates, the remeshing operation involves only a small region of polygons around the propagating crack tip. Mesh mapping technology is used to transfer the response parameters to newly generated nodes. The proposed method is validated by analysing the impact of two identical bars and the dynamic contact of crack faces in the Koyna Dam. The results show that the dynamic contact model can effectively simulate the opening and closing of crack faces, and the longer the crack is, the greater the influence of the non-linearity of the contact along a crack face on the dynamic stress intensity factor and dynamic response. Finally, a dynamic crack propagation problem is simulated. Compared with the dynamic crack response without considering the contact constraint, when considering the contact constraint, the crack initiation angle is closer to the corresponding theoretical solution, and the crack propagation path is closer to the corresponding experimental result. Thus, the dynamic contact constraint condition of crack faces has an important influence on crack propagation.

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.

Intersection-distribution-based remapping between arbitrary meshes for staggered multi-material arbitrary Lagrangian-Eulerian hydrodynamics

We present a new intersection-distribution-based remapping method between arbitrary polygonal meshes for indirect staggered multi-material arbitrary Lagrangian-Eulerian hydrodynamics. All cell-centered material quantities are conservatively remapped using intersections between the Lagrangian (old, source) mesh and the rezoned (new, target) mesh. The new nodal masses are obtained by conservative distribution of all material masses in each new cell to the cell's corners and then collecting those corner masses at new nodes. This distribution is done using a local constrained optimization approach for each cell in the new mesh. In order to remap nodal momentum we first define cell-centered momentum for each cell in the old mesh, conservatively remap this to the new mesh and then conservatively distribute the new zonal momentum to each cell's bounding nodes, again using local constrained optimization. Our method also conserves total energy by applying a new nodal kinetic energy correction that relies on a process similar to that used for remapping nodal mass and momentum. Cell-centered kinetic energy is computed, conservatively remapped and then distributed to nodes. The discrepancy between this conservatively remapped and actual nodal kinetic energy is then conservatively distributed to the internal energies of the materials in the cells surrounding each node. Unlike conventional cell-based corrections of this type, this new nodal kinetic energy correction has not been observed to drive material internal energy negative in any of our testing. Unlike flux based remapping, our new intersection-distribution method can be applied to remapping between source and target meshes that are arbitrarily different, which provides superior flexibility in the rezoning strategy. Our method is accurate, essentially conservative and essentially bounds preserving.

A Least Square Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form

This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator. Naturally, the resulting linear system is symmetric and positive definite, and thus the algorithm is easy to implement and analyze. Convergence analysis in the H2 equivalent norm is established on an arbitrary shape regular polygonal mesh. A superconvergence result is proved when the coefficient matrix is constant or piecewise constant. Numerical examples are performed which not only verify the theoretical results but also reveal some unexpected superconvergence phenomena.

Nonlinear Finite Volume Scheme Preserving Positivity for 2D Convection-Diffusion Equations on Polygonal Meshes

In this paper, a nonlinear finite volume scheme preserving positivity for solving 2D steady convection-diffusion equation on arbitrary convex polygonal meshes is proposed. First, the nonlinear positivity-preserving finite volume scheme is developed. Then, in order to avoid the computed solution beyond the upper bound, the cell-centered unknowns and auxiliary unknowns on the cell-edge are corrected. We prove that the present scheme can avoid the numerical solution beyond the upper bound. Our scheme is locally conservative and has only cell-centered unknowns. Numerical results show that our scheme preserves the above conclusion and has second-order accuracy for solution.

Mesh smoothing algorithm based on exterior angles split.

Since meshes of poor quality give rise to low accuracy in finite element analysis and kinds of inconveniences in many other applications, mesh smoothing is widely used as an essential technique for the improvement of mesh quality. With respect to this issue, the main contribution of this paper is that a novel mesh smoothing method based on an exterior-angle-split process is proposed. The proposed method contains three main stages: the first stage is independent element geometric transformation performed by exterior-angle-split operations, treating elements unconnected; the second stage is to offset scaling and displacement induced by element transformation; the third stage is to determine the final positions of nodes with a weighted strategy. Theoretical proof describes the regularity of this method and many numerical experiments illustrate its convergence. Not only is this method applicable for triangular mesh, but also can be naturally extended to arbitrary polygonal surface mesh. Quality improvements of demonstrations on triangular and quadrilateral meshes show the effectiveness of this method.

Extended virtual element method for the torsion problem of cracked prismatic beams

In this paper, we investigate the capability of the recently proposed extended virtual element method (X-VEM) to efficiently and accurately solve the problem of a cracked prismatic beam under pure torsion, mathematically described by the Poisson equation in terms of a scalar stress function. This problem is representative of a wide class of elliptic problems for which classic finite element approximations tend to converge poorly, due to the presence of singularities. The X-VEM is a stabilized Galerkin formulation on arbitrary polygonal meshes derived from the virtual element method (VEM) by augmenting the standard virtual element space with an additional contribution that consists of the product of virtual nodal basis functions with a suitable enrichment function. In addition, an extended projector that maps functions lying in the extended virtual element space onto linear polynomials and the enrichment function is employed. Convergence of the method on both quadrilateral and polygonal meshes for the cracked beam torsion problem is studied by means of numerical experiments. The computed results affirm the sound accuracy of the method and demonstrate a significantly improved convergence rate, both in terms of energy and stress intensity factor, when compared to standard finite element method and VEM.

A coupled volume-of-fluid and level-set method (VOSET) for capturing interface of two-phase flows in arbitrary polygon grid

This article presents an extension of coupled volume-of-fluid and level-set method (VOSET) for simulating free surfaces flows in arbitrary 2D polygon meshes. A series of techniques are introduced for geometric calculations in convex polygons. Newton iteration is adopted for the interface reconstruction in polygons, and incremental remapping approach is employed for the propagation of the volume fractions. The interface tracking test results suggested a second order accuracy in mixed and hexagonal grids. For the validation purpose, a Rayleigh-Taylor instability problem, a liquid column collapse problem and a single bubble rising problem were numerically studied, and the obtained results show excellent agreements with experimental data and benchmark solutions in literatures. Finally the proposed VOSET method was applied in simulating the working process of a flow-focusing microfluidic droplet generator.

An equal-order mixed polygonal finite element for two-dimensional incompressible Stokes flows

Abstract In this study, a novel equal-order mixed polygonal finite element is developed to solve the incompressible fluid problems. Our developed element, so-called Pe1Pe1, is established by the system of polygonal basis functions for both fluid pressure and velocity fields. The superiority of the present formulation over the existent low-order polygonal element is its full applicability in all arbitrary polygonal mesh families. Furthermore, our element provides a better approximation solution for pressure field in fluid analysis. To avoid the well-known instability in equal-order mixed discretizations, we develop an adequate stabilization method, which is extremely effective, yet simple, to improve the numerical performance of our polygonal finite element. In this research, the quality of the present approach with respect to the convergence rate and its stability is presented by a variety of numerical samples of the incompressible Stokes flow problems.

Virtual element method for semilinear elliptic problems on polygonal meshes

In this paper, the virtual element method is employed to approximate semilinear elliptic problems over arbitrary polygonal meshes. The nonlinear load term is approximated by employing the orthogonal L2 projection. The finite dimensional formulation and its implementation are discussed in detail and optimal a priori error estimate in H1 norm is derived. Further, two numerical experiments are conducted in order to illustrate the performance of the proposed scheme and to numerically justify the theoretical convergence rate. It is observed that the proposed method yields optimal convergence rates in both L2 and H1 norm.

A new Green–Gauss reconstruction on unstructured meshes. Part I: Gradient reconstruction

We describe a new and simple strategy based on the Gauss divergence theorem for obtaining centroidal gradients on unstructured meshes. Unlike the standard Green–Gauss (SGG) reconstruction which requires face values of quantities whose gradients are sought, the proposed approach reconstructs the gradients using the normal derivative(s) at the faces. The new strategy, referred to as the Modified Green–Gauss (MGG) reconstruction results in consistent gradients which are at least first-order accurate on arbitrary polygonal meshes. We show that the MGG reconstruction is linearity preserving independent of the mesh topology and retains the consistent behaviour of gradients even on meshes with large curvature and high aspect ratios. The gradient accuracy in MGG reconstruction depends on the accuracy of discretisation of the normal derivatives at faces and this necessitates an iterative approach for gradient computation on non-orthogonal meshes. Numerical studies on different mesh topologies demonstrate that MGG reconstruction gives accurate and consistent gradients on non-orthogonal meshes, with the number of iterations proportional to the extent of non-orthogonality. The MGG reconstruction is found to be consistent even on meshes with large aspect ratio and curvature with the errors being lesser than those from linear least-squares reconstruction. A non-iterative strategy in conjunction with MGG reconstruction is proposed for gradient computations in finite volume simulations that achieves the accuracy and robustness of MGG reconstruction at a cost equivalent to that of SGG reconstruction. The efficacy of this strategy for fluid flow problems is demonstrated through numerical investigations in both incompressible and compressible regimes. The MGG reconstruction may, therefore, be viewed as a novel and promising blend of least-squares and Green–Gauss based approaches which can be implemented with little effort in open-source finite-volume solvers and legacy codes.