Polygonal Finite Element Research Articles

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.

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

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

An n‐sided polygonal finite element for nonlocal nonlinear analysis of plates and laminates

SummaryIn this study, a locking‐free n‐sided C1 polygonal finite element is presented for nonlinear analysis of laminated plates. The plate kinematics is based on Reddy's third‐order shear deformation theory (TSDT). The in‐plane displacements are approximated using barycentric form of Lagrange shape functions. The weak‐form Galerkin formulation based on the kinematics of TSDT requires the C1 approximation of the transverse displacement over the polygonal element. This is achieved by embedding the C0 Lagrange interpolants over a cubic Bernstein‐Bezier patch defined over the n‐sided polygonal element. Such an approach ensures the continuity of the derivative field at the inter‐element edges. In addition, Eringen's stress‐gradient nonlocal constitutive equations are used in the present formulation to account for nonlocality. The effect of geometric nonlinearity is taken by considering the von Kármán geometric nonlinearity. Examples are presented to show the effect of nonlocality, geometric nonlinearity, and the lamination scheme on the bending behavior of laminated composite plates. The results are compared with analytical solutions, conventional FEM results, and with those available in the literature. Shear locking is addressed considering reduced integration and consistent interpolation techniques. The patch test is used to check the convergence of the element developed.

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

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

Numerical modeling of rock failure under dynamic loading with polygonal elements

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

Micromechanically motivated constitutive model embedded in two-dimensional polygonal finite element framework for magnetostrictive actuators

A two-dimensional constitutive model based on micromechanical domain rotation events is presented in this work to demonstrate the nonlinear actuator behavior of magnetostrictive materials, in particular, Galfenol. The model constructed upon thermodynamic principles accounts back fields which resist or aid the domain rotation events inside a grain due to external magnetomechanical loading. The developed model is then incorporated into the polygonal finite element technique that combines Voronoi-based discretization with the hybrid finite element method. In this approach, the stress and magnetic flux density are treated as approximate functions inside the element, but the mechanical displacement and magnetic potential, which act as degrees of freedom, are defined only along the element boundary. This approach allows each randomly generated Voronoi polygon in the plane discretization to act as a single finite element mimicking an individual magnetomechanical grain in a polycrystalline Galfenol, eliminating the need for further subdiscretization of the Voronoi polygon. This coupled framework simulates the nonlinear actuator characteristics of the magnetostrictive material under complex magnetomechanical loading conditions in line with the experimental observations reported in the literature.

Active vibration control of GPLs-reinforced FG metal foam plates with piezoelectric sensor and actuator layers

This paper investigates free vibration and dynamic responses of smart FG metal foam plate structures reinforced by graphene platelets (GPLs). We then analyze active control of FG metal foam plates with piezoelectric sensor and actuator layers. To provide numerical solution of underlying problems, we develop a computational approach based on a generalized C0-type higher-order shear deformation theory (C0-HSDT) polygonal finite element formulation (PFEM), which is suitable for modeling both thick and thin plates. To enhance accuracy of solution, we use in PFEM quadratic serendipity shape functions in combination with a generalized C0-HSDT. The FG core layers are constituted by combining between two porosity distributions and three GPL dispersion patterns distributed along the plate's thickness while two piezoelectric layers are perfectly bonded on the both bottom and top surfaces of host plate. The mechanical displacement field is approximated based on C0-HSDT while the electric potential distribution through the thickness for each piezoelectric layer is assumed to be a linear function. For active control, a constant velocity feedback scheme is employed through a closed loop control with piezoelectric sensors and actuators. The effect of the porosity coefficient, weight fraction of GPLs on the plate's behaviors with various porosity distributions and GPL dispersion patterns are evidently demonstrated through numerical examples.

On structural topology optimization considering material nonlinearity: Plane strain versus plane stress solutions

We present a structural topology optimization framework considering material nonlinearity by means of a tailored hyperelastic formulation. The nonlinearity is incorporated through a hyperelastic constitutive model, which is capable of capturing a range of nonlinear material behavior under both plane strain and plane stress conditions. We explore both standard (i.e. quadrilateral) and polygonal finite elements in the solution process, and achieve smooth convergence in both the optimization process and the solution of nonlinear state equations. Numerical examples are presented, which demonstrate that the topology optimization framework can effectively capture the influence of various material behaviors, load levels and loading conditions (i.e. plane stress versus plane strain) on the optimal topologies.

An extended polygonal finite element method for large deformation fracture analysis

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

Auxetic structure design using compliant mechanisms: A topology optimization approach with polygonal finite elements

Compliant mechanisms are monolithic structures where the movement is given by the flexibility of the structure rather than the presence of joints and pins. The absence of joints allows the construction of compliant mechanisms in microscale. In this work, the compliant mechanism is designed by using topology optimization to generate microstructure unit cells that simulate the effect of auxetic materials, i.e. those with negative Poisson's ratio. Polygonal finite element meshes are introduced in the topology optimization formulation to avoid the hinges (one-node connections) in the compliant mechanism design. A pattern repetition constraint is applied to generate auxetic macrostructures. An integrated approach that combines a projection technique with a mapping technique is adopted to include the minimum member size constraint, making the topology optimization results possible to be manufactured. The connection of the topology optimization approach with additive manufacturing is demonstrated using 3D printers based on FFF (fused filament fabrication) and PolyJet technologies. Thus, computational simulation in connection with rapid prototyping are carried out to verify the results.

A finite element formulation in boundary representation for the analysis of nonlinear problems in solid mechanics

The contribution is concerned with a numerical element formulation in boundary representation. It results in a polynomial element description with an arbitrary number of nodes on the boundary. Scaling the boundary description determines the interior domain. The scaling approach is adopted from the so-called scaled boundary finite element method (SBFEM), which is a semi-analytical formulation to analyze problems in linear elasticity. Within this method, the basic idea is to scale the boundary with respect to a scaling center. The boundary, which is denoted as circumferential direction, and the scaling direction span the parameter space. In the present approach, interpolations in scaling direction and circumferential direction are introduced. The interpolation in circumferential direction is independent of the scaling direction. The formulation is suitable to analyze problems in nonlinear solid mechanics. The displacement degrees of freedom are located at the nodes on the boundary and in the interior element domain. The degrees of freedom located at the interior domain are eliminated by static condensation, which leads to a polygonal finite element formulation with an arbitrary number of nodes on the boundary. The element formulation allows per definition for Voronoi meshes and quadtree mesh generation. Numerical examples give rise to the performance of the present approach in comparison to other polygonal element formulations, like the virtual element method (VEM). Some benchmark tests show the capability of the element formulation. A comparison to standard and mixed element formulations is presented. The present approach is perfectly suitable to model heterogeneous structures with inclusions and voids. It avoids also staircase approximation of curved boundaries.

Multi-material topology optimization for additive manufacturing using polytree-based adaptive polygonal finite elements

Abstract The development of additive construction, which is an automated production process based layer-by-layer control, opens a new era for building design. It creates opportunities for novel shapes that would not have been technologically feasible in the past. The key to this revolution resides in the link between material development and computational-based optimization design tools to produce optimized structures. Topology optimization is an intelligent approach to seek the best design in many fields of computational mechanics. By analysing the information inside the determined domain, it changes the topology of such domain into a new design to satisfy the given criterion. In most of the work, ideal materials are assumed to implement the topology optimization problems. Therefore, the optimal designs are still considered as reference or theoretical solutions while the practical problems need more than that. The multi-material topology optimization (MMTOP) is established to evaluate the behaviour of many materials in a problem and give an applicable solution for design. In this study, we present a polytree-based adaptive methodology for multi-material topology optimization (MMTOP). Polytree data structure is introduced as a general recursive multi-level mesh that is automatically refined in processing based on error analysis. A new definition of filter radius is also proposed to improve the efficiency of filters and optimized results. The combination of polytree meshes and adaptive filters not only clarifies the interfaces between material phases (including void phase), but also decreases the computing time of the overall process in comparison to using the regular fine meshes. Several benchmark and practical problems are considered to show distinct features of the proposed method. 3D printed prototypes are presented to demonstrate the fabrication feasibility of this method.

A finite element formulation in boundary representation for the analysis of nonlinear problems in solid mechanics

AbstractThe contribution is concerned with a numerical element formulation for the analysis of the nonlinear behavior in solid mechanics. It is based on the so‐called scaled boundary finite element method (SB‐FEM). The SB‐FEM is as a semi‐analytical formulation to analyze problems of linear elasticity. The basic idea is to scale the boundary with respect to a scaling center. Hence, the boundary denoted as circumferential direction and the scaling direction span the parameter space. The original SB‐FEM approach is based on an analytical ansatz for the displacement response preserving equilibrium in scaling direction. In the present approach, an interpolation in scaling direction is introduced to account for nonlinear problems. Lagrange interpolations with different polynomial order are therefore discussed. The interpolation in circumferential direction is independent of the scaling direction and is defined by independent Lagrange functions. The displacement degrees of freedom are located at the nodes on the boundary and in the interior element domain. The degrees of freedom located in the interior of the domain are eliminated by static condensation, which leads to a polygonal finite element formulation with an arbitrary number of nodes on the boundary. The element formulation allows a priori for Voronoi meshes and quadtree mesh generation. Numerical examples give rise to the performance of the present approach in comparison to other element formulations, like mixed formulations.

Geometrically nonlinear polygonal finite element analysis of functionally graded porous plates

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

Finite deformation cohesive polygonal finite elements for modeling pervasive fracture

We introduce a framework for modeling dynamic fracture problems using cohesive polygonal finite elements. Random polygonal meshes provide a robust, efficient method for generating an unbiased network of fracture surfaces. Further, these meshes have more facets per element than standard triangle or quadrilateral meshes, providing more possible facets per element to insert cohesive surfaces. This property of polygonal meshes is advantageous for the modeling of pervasive fracture. We use both Wachspress and maximum entropy shape functions to form a finite element basis over the polygons. Fracture surfaces are captured through dynamically inserted cohesive zone elements at facets between the polygons in the mesh. Contact is enforced through a penalty method that is applied to both closed cohesive surfaces and general interpenetration of two polygonal elements. Several numerical examples are presented that illustrate the capabilities of the method and demonstrate convergence of solutions.

PolyMat: an efficient Matlab code for multi-material topology optimization

We present a Matlab implementation of topology optimization for compliance minimization on unstructured polygonal finite element meshes that efficiently accommodates many materials and many volume constraints. Leveraging the modular structure of the educational code, PolyTop, we extend it to the multi-material version, PolyMat, with only a few modifications. First, a design variable for each candidate material is defined in each finite element. Next, we couple a Discrete Material Optimization interpolation with the existing penalization and introduce a new parameter such that we can employ continuation and smoothly transition from a convex problem without any penalization to a non-convex problem in which material mixing and intermediate densities are penalized. Mixing that remains due to the density filter operation is eliminated via continuation on the filter radius. To accommodate flexibility in the volume constraint definition, the constraint function is modified to compute multiple volume constraints and the design variable update is modified in accordance with the Zhang-Paulino-Ramos Jr. (ZPR) update scheme, which updates the design variables associated with each constraint independently. The formulation allows for volume constraints controlling any subset of the design variables, i.e., they can be defined globally or locally for any subset of the candidate materials. Borrowing ideas for mesh generation on complex domains from PolyMesher, we determine which design variables are associated with each local constraint of arbitrary geometry. A number of examples are presented to demonstrate the many material capability, the flexibility of the volume constraint definition, the ease with which we can accommodate passive regions, and how we may use local constraints to break symmetries or achieve graded geometries.

N-sided polygonal hybrid finite elements with unified fundamental solution kernels for topology optimization

• A model of n -sided polygonal hybrid FE based- topology optimization is presented. • Only line integrals along the element boundary are involved in the computation. • Easy to construct any polygons for adapting complex design domains. • The present model n -sided polygonal hybrid FE is with high accuracy. In topology optimization, the optimized design can be obtained based on spatial discretization of design domain using natural polygonal finite elements to reduce the influence of mesh geometry on topology optimization solutions. However, the natural polygonal finite elements require separate interpolants for each type of elements and involve troublesome domain integrals. In this study, an alternative n -sided polygonal hybrid finite element possessing multiple-node connection is formulated in a unified form to compress the checkerboard patterns caused by numerical instability in topology optimization. Different from the natural polygonal finite elements, the present polygonal hybrid finite elements involve two sets of independent displacement fields. The intra-element displacement field defined inside the element is approximated by the linear combination of the fundamental solution of the problem to achieve the purpose of the local satisfaction of the governing equations of the problem, but not the specific boundary conditions and the inter-element continuity conditions . To overcome such drawback, the inter-element displacement field defined over the entire element boundary is independently approximated by means of the conventional shape function interpolation. As a result, only line integrals along the element boundary are involved in the computation, whose dimension is reduced by one compared to the domain integrals in the natural polygonal finite elements, and more importantly, allowing us to flexibly construct any polygons from Voronoi tessellations in discretizing complex design domains using same fundamental solution kernels. Numerical results obtained indicate that the present n -sided polygonal hybrid finite elements can produce more accurate displacement solutions and smaller mean compliance, compared to the standard finite elements and the natural polygonal finite elements.

Optimal Numerical Integration Schemes for a Family of Polygonal Finite Elements with Schwarz–Christoffel Conformal Mapping

This paper presents optimal numerical integration schemes for a family of Polygonal Finite Elements with Schwarz–Christoffel (SC) conformal mapping with improved convergence properties. With Wachspress interpolation, the arbitrary polygons of a finite element mesh are first mapped to regular polygons using isoparametric mapping and then to the unit-disk using SC mapping. Numerical integration is done over this unit-disk by determining integration points which are optimal with respect to Frobenius norm of stiffness matrix and infinity norm of displacement field. Lowest order Gauss quadrature points on isoparametrically mapped domain are considered as initial conditions for optimization. Hence this method achieves better accuracy and faster convergence compared to other contemporary methods with fewer computational resources. The method automatically reduces to standard Gauss quadrature-based integration when polygons are quadrilaterals as a special case. In general, only the element boundary poses C0 continuity whereas the element interior has higher order spectral properties. Numerical results presented for a few benchmark problems show good accuracy of the proposed schemes. The method is validated considering structured as well as unstructured meshes consisting of pentagons and hexagons which is a significant step forward to meshing and solving elasticity problem in complicated geometry and interfaces, material microstructure, and so forth.

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

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

A simple adaptive mesh refinement scheme for topology optimization using polygonal meshes

The objective of topology optimization is to find the most efficient distribution of material (optimal topology) in a given domain, subjected to design constraints defined by the user. The quality of the new boundary representation depends on the level of mesh refinement: The greater the number of elements in the mesh, the better is the representation of the optimized structure. However, the use of refined meshes implies a high computational cost, particularly with regard to the numerical solution of the linear systems of equations that arise from the finite element method. This paper proposes a new computational strategy for adaptive local mesh refinement using polygonal finite elements in arbitrary two-dimensional domains. The idea is to perform a mesh refinement in regions of material concentration, mostly in inner and outer boundaries, and a mesh derefinement in regions of low material concentration, such as in internal holes. Thus, it is possible to obtain high-resolution optimal topologies with a relatively low computational cost. Representative examples are presented to demonstrate the robustness and efficiency of the proposed methodology by comparing the results obtained here with results from the literature where refined meshes are kept constant throughout the topology optimization process.