Virtual Element Research Articles

An enriched virtual element method for 2D‐3C generalized membrane shell model on surface

AbstractDealing with complex shell surfaces using the finite element method, we are often limited to simple geometric meshes such as triangles and quadrangles and have to refine the meshes to meet the calculation accuracy requirements, significantly increasing the calculation cost. The virtual element method (VEM), a new numerical method with high mesh flexibility, has been widely applied to many physical and mechanical problems. To the best of our knowledge, this method has yet to be studied for membrane shell models so far. In this paper, for the first time, we provide an enriched conforming VEM discrete scheme for the two‐dimensional three‐component (2D‐3C) generalized membrane shell (GMS) model proposed by Ciarlet et al. Furthermore, we prove the VEM discrete solution's existence and uniqueness for the GMS. Numerical experiments demonstrate that this discrete scheme is convergent and stable and can accurately represent the membrane stress state of conical, cylindrical, hyperbolical, and spherical shells clamped along a portion of their lateral faces. At the same time, we show the diversity of the grid subdivision. Thus, we develop the VEM for the GMS model successfully. In the future, we will continue to study the VEM for other shell models.

BDDC Preconditioners for Virtual Element Approximations of the Three-Dimensional Stokes Equations

BDDC Preconditioners for Virtual Element Approximations of the Three-Dimensional Stokes Equations

An enhanced corotational Virtual Element Method for large displacements in plane elasticity

An enhanced virtual element formulation for large displacement analyses is presented. Relying on the corotational approach, the nonlinear geometric effects are introduced by assuming nodal large displacements but small strains in the element. The element deformable behavior is analyzed with reference to the local system, corotating with the element during its motion. Then, the large displacement-induced nonlinearity is accounted for through the transformation matrices relating the local and global quantities. At the local level, the Virtual Element Method is adopted, proposing an enhanced procedure for strain interpolation within the element. The reliability of the proposed approach is explored through several benchmark tests by comparing the results with those evaluated by standard virtual elements, finite element formulations, and analytical solutions. The results prove that: (i) the corotational formulation can be efficiently used within the virtual element framework to account for geometric nonlinearity in the presence of large displacements and small strains; (ii) the adoption of enhanced polynomial approximation for the strain field in the virtual element avoids, in many cases, the need for ad-hoc stabilization procedures also in the nonlinear geometric framework.

An extended virtual element method for elliptic interface problems

This paper proposes an extended virtual element method based on the combination of the extended finite element method and the virtual element method on interface-unfitted meshes for solving elliptic interface problems. The well-posedness of the discrete problem is shown and the optimal error estimate is derived, which is independent of the location of the interface relative to the mesh and the contrast between the diffusion coefficients. Numerical experiments verify the theoretical results.

The Hermite-type virtual element method with interior penalty for the fourth-order elliptic problem

The Hermite-type virtual element method with interior penalty for the fourth-order elliptic problem

Hydrogen assisted cracking using an efficient virtual element scheme

In this paper, a novel virtual element formulation is proposed for phase field modeling of hydrogen assisted cracking (HAC). The multiphysics variational framework is decoupled into three parts which can be solved in a staggered manner, namely, the mechanical, diffusion, and damage sub-problems. The damage and diffusion sub-problems are treated as reaction–diffusion types of equations. The former is subjected to irreversibility constraints while the latter incorporates a transport term to represent the effect of mechanical loading. An efficient super-convergent patch recovery (SPR) scheme is utilized to determine the coefficient of transport term in the hydrogen diffusion equation. Several qualitative and quantitative benchmark problems are considered to validate the proposed framework’s performance. The results show excellent agreement with the corresponding FEM and experimental studies. Meanwhile, without loss of numerical accuracy, the VEM outperforms the FEM with regards to CPU time & memory efficiency and exhibits remarkable versatility in the selection of different element types for simulation. Moreover, this work demonstrates the potential of facilitating the VEM to solve real-scale fracture mechanics problems with multi-physics coupling.

Homogenization of composite materials reinforced with unidirectional fibres with complex curvilinear cross section: a virtual element approach

<p>The paper presents an augmented curvilinear virtual element method to determine homogenized in-plane shear material moduli of long-fibre reinforced composites in the framework of asymptotic homogenization method. The new virtual element combine an exact representation of the curvilinear computational geometry for complex fibre cross section shapes through an innovative two-dimensional cubature suite for NURBS-like polygonal domains. A selection of representative numerical tests supports the accuracy and efficiency of the proposed approach for both doubly periodic and random fibre arrangement with matrix domain.</p>

SUPG-stabilized stabilization-free VEM: a numerical investigation

<abstract><p>We numerically investigate the possibility of defining Stabilization-Free Virtual Element discretizations–i.e., Virtual Element Method discretizations without an additional non-polynomial non-operator-preserving stabilization term–of advection-diffusion problems in the advection-dominated regime, considering a Streamline Upwind Petrov-Galerkin stabilized formulation of the scheme. We present numerical tests that assess the robustness of the proposed scheme and compare it with a standard Virtual Element Method.</p></abstract>

Numerical recipes of virtual element method for phase field modeling of brittle fracture

In this work, a new and efficient virtual element formulation for non-standard phase field model of brittle fracture is presented. A multi-pass alternative minimization solution scheme based on algorithm operator splitting is utilized, which decouples the whole problem into two parts, namely, mechanical and damage sub-problems. The former is treated as elasto-static problem, while the latter one is treated as Poisson-type of reaction-diffusion equation subjected to bounded and irreversibility constraint. To demonstrate the performance of proposed formulation, several benchmark problems are studied and results are in good agreement with corresponding finite element calculations and experimental studies.

Proportional Topology Optimization algorithm with virtual elements for multi-material problems considering mass and cost constraints

This paper presents an extension of the Proportional Topology Optimization (PTO) with virtual elements for multi-material problems with mass and cost constraints. In particular, the linear virtual element method (VEM) is constructed on unstructured polygonal meshes. The linear VEM is desirable in the sense that numerical integration is not explicitly required, significantly reducing the computational effort. Furthermore, the unstructured polygonal mesh naturally eliminates the issue of one-node connections encountered by the usual quadrilateral mesh. A feature of PTO is that it does not require sensitivity information, i.e., the derivative of the objective function with respect to design variables. Instead, the amount of material distributed into each element is determined proportionally to the contribution of that element to the objective function. For multi-material problems, the Ordered Solid Isotropic Material with Penalization (Ordered SIMP) technique is integrated into the PTO framework. Compared to other techniques for problems that involve multiple materials, Ordered SIMP has the advantage that computational cost does not depend on the number of materials. Furthermore, for the first time, the PTO approach is extended to consider two types of constraints: mass and cost simultaneously. The feasibility and efficiency of the proposed method are demonstrated via several benchmark examples and comparisons with the existing approach.

Augmented Reality in Read Aloud: An English Language Interactive Engagement

Currently, Augmented Reality (AR) technology is emerging as a popular means to encourage children to enjoy reading books. AR technology, which creates a virtual element from a real physical environment, increases a sense of connectivity between the real and the virtual world. The Read-Aloud project aimed to spark interest in reading English language storybooks among primary school children by innovating the chosen storybook with embedded AR technology. The book, 'Nyatoh the Friendly Tree,' was selected in support of United Nations Sustainable Development Goal No. 15 on the Sustainability of Life on Land, particularly focusing on issues related to trees and sustainable forests. Primary research through a simple questionnaire was distributed to the students and analysed by the researchers using a statistical percentage table. Oral feedback and interactions were observed and recorded by the researchers. A total of 160 students, with 82 aged 12 and 78 aged 10, from a government primary school in the urban city experienced AR in the read-aloud session. The book was initially read to the students, followed by language activities and hands-on AR experiences using smartphones and physical books. All activities were conducted in smaller groups of five. The AR features 3D models of the main character and side characters, including animals from the rainforest. Comprehension questions in the form of a quiz were generated, and to expand vocabulary, audio explanations of words in animation were created. Feedback was gathered at the end of the session in two categories: (a) interest, engagement, and interaction with the story, and (b) comprehension of the language and story. Students reported positive experiences with AR in both categories and generally found the activity fun, interesting, and enjoyable and were motivated to read. While scores in the comprehension area were lower, they still indicated a positive impact of AR on vocabulary comprehension, contributing to the overall understanding of the story. Interaction with animation also plays a role in supporting the sense-making of non-verbal representation. This project demonstrated that AR holds tremendous appeal in generating interest in reading when it incorporates a modification of physical reality with virtual elements. Exploring the potential of AR in storybooks and read-aloud sessions is valuable, especially considering the benefits of reading on learners' development. The research was still in its early stages, with additional variables to be considered. Longitudinal studies could be conducted for clearer validity and reduction of the novelty effect. This includes identifying individual learning styles in software design, providing training for both teachers and students, and developing appropriate learning activities for AR applications.

Reduced basis stabilization and post-processing for the virtual element method

We present a reduced basis method for cheaply constructing (possibly rough) approximations to the nodal basis functions of the virtual element space, and propose to use such approximations for the design of the stabilization term in the virtual element method and for the post-processing of the solution.

Analysis of a stabilization-free quadrilateral Virtual Element for 2D linear elasticity in the Hu-Washizu formulation

The Virtual Element Method (VEM) for the elasticity problem is considered in the framework of the Hu-Washizu variational formulation. In particular, a couple of low-order schemes presented in [1], are studied for quadrilateral meshes. The methods under consideration avoid the need of the stabilization term typical of the VEM, due to the introduction of a suitable projection on higher-order polynomials. The schemes are proved to be stable and optimally convergent in a compressible regime, including the case where highly distorted (even non-convex) meshes are employed.

A co‐rotational virtual element method for 2D elasticity and plasticity

AbstractThe virtual element method (VEM) allows discretization of the problem domain with polygons in 2D. The polygons can have an arbitrary number of sides and can be concave or convex. These features, among others, are attractive for meshing complex geometries. VEM applied to linear elasticity problems is now well established. Nonlinear problems involving plasticity and hyperelasticity have also been explored by researchers using VEM. Clearly, techniques for extending the method to nonlinear problems are attractive. In this work, a novel first‐order consistent VEM is applied within a static co‐rotational framework. To the author's knowledge, this has not appeared before in the literature with virtual elements. The formulation allows for large displacements and large rotations in a small strain setting. For some problems avoiding the complexity of finite strains, and alternative stress measures, is warranted. Furthermore, small strain plasticity is easily incorporated. The basic method, VEM specific implementation details for co‐rotation, and representative benchmark problems are illustrated. Consequently, this research demonstrates that the co‐rotational VEM formulation successfully solves certain classes of nonlinear solid mechanics problems. The work concludes with a discussion of results for the current formulation and future research directions.

Virtual Element Method for Control Constrained Dirichlet Boundary Control Problem Governed by the Diffusion Problem

Virtual Element Method for Control Constrained Dirichlet Boundary Control Problem Governed by the Diffusion Problem

An improved integrated framework based nodal density variable and Voronoi polygon for FE-based topology optimization

In this paper, an improved integrated framework is developed to deal with the numerical instability problems in FE-based topology optimization design methods. The proposed method mainly consists of two coupled computational layers: the upper layer for material geometric description, and the lower layer for structural analysis and sensitivity calculation. In the upper layer, the design domain is discretized by Delaunay triangulation, and a smooth and continuous material density description is obtained by interpolating the densities at nodes of Delaunay triangulation with Shepard interpolants. In the lower layer, the dual Voronoi polygon for Delaunay triangulation is used to generate the finite element mesh, and the virtual element method is applied to handle Voronoi polygon for structural response. Following this framework, the separate description of density distribution avoids the problem of geometric description relying on FE mesh in some existing work. Meanwhile, the use of Voronoi polygon provides more accurate finite element discretization of constantly changing material distributions. Typical numerical examples for the minimum compliance optimization problem reveal that the proposed method can eliminate checkerboards and mesh-dependencies effectively, and get approximate discrete 0/1 solutions without any filter. The simulation results demonstrate the efficiency and robustness of the proposed formulation and numerical techniques.

A new virtual functional element method for deviation prediction of assembled structures with parallel connection chain

The assembly precision prediction of industrial products with numerous tolerance information and complex assembly relations is a challenge for existing deviation analysis methods since their dimensional chains usually form parallel connections. The parallel connection chain is often transformed into a serial connection chain by discarding some tolerance information and physical constraints in practical application, which leads to amplifying the range of predicted results and reducing the predicted accuracy. In this paper, a virtual functional element method is proposed to simplify the parallel connection chain by integrating the tolerance information into the torsor parameters of specified virtual functional elements. A general Jacobian-Torsor model is developed to solve the simplified parallel connection chain, in which the Boolean intersection operation on torsor parameters is used to obtain comprehensive deviation results for multiple branches in a parallel connection chain. Meanwhile, the locking effect and leverage activation effect are proposed to characterize the effect of the invariant degree in parallel connection chain. Finally, an assembly case of a power plant is applied to validate the proposed method, and the results show that it has a high potential to improve the prediction accuracy of assembly deviation compared to three conventional methods.

Meso-viscoelastic modeling of solid propellant using XCT technology and virtual element method

This paper proposes a scheme to develop 2D mesoscale propellant models from X-ray computed tomography (XCT) images, including digital image processing (DIP)-based models and molecular dynamics (MD)-based models. In the MD-based approach, parametric models can be constructed by presuming that the particles are discs that satisfy the size distribution and volume fraction derived from XCT images of propellant. Based on the advantages of the virtual element method (VEM) for simulating the deformation of composite materials, this study obtains statistical information from XCT images to generate mesostructural models and compare their mechanical behaviors. Under the assumption of small deformation, the mesoscopic behaviors predicted by both mesostructural models are in good agreement, indicating that simpler MD-based models are adequate to characterize the overall viscoelastic properties of propellant. The impacts of propellant mesostructure parameters were addressed by using MD-based models.

Conforming VEM for general second-order elliptic problems with rough data on polygonal meshes and its application to a Poisson inverse source problem

This paper focuses on the analysis of conforming virtual element methods (VEMs) for general second-order linear elliptic problems with rough source terms and applies it to a Poisson inverse source problem with rough measurements. For the forward problem, when the source term belongs to [Formula: see text], the right-hand side for the discrete approximation defined through polynomial projections is not meaningful even for standard conforming VEM. The modified discrete scheme in this paper introduces a novel companion operator in the context of conforming VEM and allows data in [Formula: see text]. This paper has three main contributions. The first contribution is the design of a conforming companion operator [Formula: see text] from the conforming virtual element space to the Sobolev space [Formula: see text], a modified virtual element scheme, and the a priori error estimate for the Poisson problem in the best-approximation form without data oscillations. The second contribution is the extension of the a priori analysis to general second-order elliptic problems with source term in [Formula: see text]. The third contribution is an application of the companion operator in a Poisson inverse source problem when the measurements belong to [Formula: see text]. Tikhonov’s regularization technique regularizes the ill-posed inverse problem, and the conforming VEM approximates the regularized problem given a finite measurement data. The inverse problem is also discretized using the conforming VEM and error estimates are established. Numerical tests on different polygonal meshes for general second-order problems, and for a Poisson inverse source problem with finite measurement data verify the theoretical results.

The Morley-type virtual element method for the Navier–Stokes equations in stream-function form

The nonconforming Morley-type virtual element method for the incompressible Navier–Stokes equations formulated in terms of the stream-function on simply connected polygonal domains (not necessarily convex) is designed. A rigorous analysis by using a new enriching operator is developed. More precisely, by employing such operator, we provide novel discrete Sobolev embeddings, which allow to establish the well-posedness of the discrete scheme and obtain optimal error estimates in broken H2-, H1- and L2-norms under minimal regularity condition on the weak solution. The velocity and vorticity fields are recovered via a postprocessing formulas. Furthermore, a new algorithm for pressure recovery based on a Stokes complex sequence is presented. Optimal error estimates are obtained for all the postprocessed variables. Finally, the theoretical error bounds and the good performance of the method are validated through several benchmark tests.