The Stokes Complex for Virtual Elements with Application to Navier\u2013Stokes Flows

In the present work, to describe the behavior of the shell, the classical theory based on the Kirchhoff-Love hypotheses is used. An approach is proposed which allows us to approximate model equations, taking account of any external loads. Virtual elements and expansion in Fourier's series are applied to determine the stress state of isotropic and orthotropic rotational shells under local loading conditions. Calculation results for shells subjected to different types of local mechanical loads are presented. Comparison of the stress state obtained using virtual elements and that calculated by the finite elements method is carried out. Calculations indicate that the application of the concept of a virtual shell element and the decomposition of loading in Fourier's series make it possible to reduce the dimensionality of the problem by 1.In addition, the high accuracy of approximation of external loads and the characteristics of the stress state of isotropic and orthotropic rotational shells is provided.

• Weak formulation of configurational forces (CFs) in the VEM is derived. • Automated mesh refinement based on CFs improves VEM approximations. • Accurate crack tip loading analyses are demonstrated. • Internal moments in virtual elements must not be disregarded in fracture analyses. • Efficient and accurate crack growth simulations with non-convex crack tip elements. The Virtual Element Method (VEM) constitutes a generalization of the Finite Element Method, allowing for spatial discretizations with arbitrary, even non-convex polygonal elements. This offers significant advantages, particularly for problems involving moving discontinuities, local stress concentrations, or multiple phases with distinctly different dimensions. The theory of configurational forces (CF) represents a very flexible basis to quantify driving forces for virtual defect displacements within a thermodynamical framework, and at the same time provides an indicator for the local mesh quality in numerical calculations. In combination, the VEM and the CF theory thus constitute an effective basis for numerical simulations of crack growth within the context of a classical approach with strong discontinuities and sharp crack tips. Challenges to cope with are attributable to the lacking knowledge of shape functions within virtual elements, the numerical integration on arbitrary polygonal elements, and the requirement of an automated generation of polygonal meshes. The weak form of the energy-momentum balance is derived from a variational approach, uniquely separating nodal CFs acting at free surfaces from those driving the defect. Furthermore, internal degrees of freedom, which are inherent to higher-order virtual elements, have to be interpreted within the context of CFs. Unphysical, so-called spurious, CFs are exploited for an improved mesh design at crack tips, eventually generating accurate results of crack tip loading and crack path simulation.

For a high speed mobile communication system, Doppler shift affects its performance seriously. In the future, broad band communication based on orthogonal frequency division multiplexing which depends on the orthogonality among sub-carriers will become popular. The absence of the orthogonality due to being destroyed by Doppler shift, leads to the failure of signal demodulation. So Doppler shift must be estimated and compensated for, which is the main purpose of previous work. On the other hand, many applications have shown that Doppler shift can be utilized to acquire the direction and speed or improve the quality of a signal. In this paper, we propose a method of not only estimating and compensating for Doppler shift, but also generating multiple non-frequency shifted signals, which can be regarded as the output of a virtual antenna array. As to the method, uniform phase sampling is the key algorithm. At first, the relation between uniform time sampling and uniform phase sampling is discussed in detail. This relation shows that the equivalence between uniform phase sampling and uniform time sampling is the necessary and sufficient condition for a non Doppler shifted signal. Next, the algorithm of Doppler shift compensation and virtualized antenna array is proposed, in which 1) original Doppler shifted signal is processed with interpolation, 2) new signals are generated by uniform phase sampling and buffered, 3) buffered new signals are read out by uniform time sampling. The theory of this process and the performance improvement for a high speed mobile communications system is mathematically analyzed, and the hardware architecture model of this algorithm is also given. The diversity gain could be obtained when an antenna array is used. In order to verify that this virtualized antenna array has the same benefit, the ability to suppress the interference and the bit error rate is analyzed with numerical simulation. The number of virtual elements and the virtual element distance are two variables related to the direction pattern of virtual antenna array. The effects of these two variables are given by the simulation, showing that the more virtual elements, the narrower beam are obtained. But more virtual elements result in more complicated hardware source. In addition, the communications scenarios of two communications radiators at different sites are simulated to verify whether this algorithm can suppress interference signal. The frequency spectrum of beamformed virtual antenna array signal shows that the interference signal can be suppressed effectively. These characteristics cannot be provided by pure Doppler frequency shift compensation. Thus these results show that high speed mobile communication systems on aircrafts or high speed trains would obtain better performances when a received Doppler shift signal is processed by this method to construct a virtual antenna array.

In this paper, we tackle the problem of constructing conforming Virtual Element spaces on polygons with curved edges. Unlike previous VEM approaches for curvilinear elements, the present construction ensures that the local VEM spaces contain all the polynomials of a given degree, thus providing the full satisfaction of the patch test. Moreover, unlike standard isoparametric FEM, this approach allows to deal with curved edges at an intermediate scale, between the small scale (treatable by homogenization) and the bigger one (where a finer mesh would make the curve flatter and flatter). The proposed method is supported by theoretical analysis and numerical tests.

A variationally consistent numerical approach based on the Virtual Element Method (VEM) is presented for the analysis of 2D elastoplasticity problems. The mixed Hu-Washizu functional of elasticity is extended to incorporate the energy contributions specific to the finite-step elastoplastic problem. It is demonstrated how the governing equations of the discretized elastoplastic problem - including the loading-unloading conditions - emerge naturally as the stationarity conditions of the VEM-discretized functional. Spurious hourglass modes are prevented by formulating a self-stabilized version of Virtual Elements (VEs) that exploits the possibility offered by the mixed approach to define strain and displacement approximations of the same order. The insensitivity of VEs to element distortion and the possibility to use polygonal elements with any shape and number of edges is tested with the analysis of several benchmarks from the literature. It is shown how accurate solutions can be obtained also in the case of non-convex quadrilateral or pentagonal elements. Additionally, the role of internal moment degrees of freedom in preventing elastoplastic locking at the plastic failure limit is elucidated.

This paper presents stochastic virtual element methods for propagating uncertainty in linear elastic stochastic problems. We first derive stochastic virtual element equations for 2D and 3D linear elastic problems that may involve uncertainties in material properties, external forces, boundary conditions, etc. A stochastic virtual element space that couples the deterministic virtual element space and the stochastic space is constructed for this purpose and used to approximate the unknown stochastic solution. Two numerical frameworks are then developed to solve the derived stochastic virtual element equations, including a Polynomial Chaos approximation based approach and a weakly intrusive approximation based approach. In the Polynomial Chaos based framework, the stochastic solution is approximated using the Polynomial Chaos basis and solved via an augmented deterministic virtual element equation that is generated by applying the stochastic Galerkin procedure to the original stochastic virtual element equation. In the weakly intrusive approximation based framework, the stochastic solution is approximated by a summation of a set of products of random variables and deterministic vectors, where the deterministic vectors are solved via converting the original stochastic problem to deterministic virtual element equations by the stochastic Galerkin approach, and the random variables are solved via converting the original stochastic problem to one-dimensional stochastic algebraic equations by the classical Galerkin procedure. This method avoids the curse of dimensionality in high-dimensional stochastic problems successfully since all random inputs are embedded into one-dimensional stochastic algebraic equations whose computational effort weakly depends on the stochastic dimension. Numerical results on 2D and 3D problems with low- and high-dimensional random inputs demonstrate the good performance of the proposed methods.

The stabilized nonconforming virtual element method for the Darcy–Stokes problem

We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.

Virtual elements and zero thickness interface-based approach for fracture analysis of heterogeneous materials

The thesis deals with theoretical and applicative aspects of some innovative numerical techniques for the simulation of the flow in Discrete Fracture Networks (DFN). In particular, the recently developed Virtual Element Method (VEM) is considered. A VEM-SUPG stabilized formulation for advection-diffusion problems is defined and studied theoretically and numerically, as well as a residual a posteriori error estimate which does not include any term depending on the VEM stabilization form. Regarding DFN flow simulations, an approach based on Virtual Elements and standard domain decomposition techniques such as Mortar methods is introduced and studied, also in combination with the use of orthogonal polynomials to avoid numerical instabilities that arise when computing polynomial projections on very badly shaped elements. Finally, we consider a constrained optimization formulation of the problem of computing the flow in DFNs and we develop a residual based a posteriori error estimate that contains non standard terms related to the geometrical non-conformity of the mesh on each fracture to the intersections between them.

The geometric shape of an element plays a key role in computational methods. Triangular and quadrilateral shaped elements are utilized by standard finite element methods. The pioneering work of Wachspress laid the foundation for polygonal interpolants which introduced polygonal elements. Tessellations may be considered as the next stage of element shape evolution. In this work, we investigate the topology optimization of tessellations as a means to coalesce art and engineering. We mainly focus on M.C. Escher's tessellations using recognizable figures. To solve the state equation, we utilize a Mimetic Finite Difference inspired approach, known as the Virtual Element Method. In this approach, the stiffness matrix is constructed in such a way that the displacement patch test is passed exactly in order to ensure optimum numerical convergence rates. Prior to exploring the artistic aspects of topology optimization designs, numerical verification studies such as the displacement patch test and shear loaded cantilever beam bending problem are conducted to demonstrate the accuracy of the present approach in two-dimensions.

Structural damage detection based on virtual element boundary measurement

The virtual element method (VEM) was developed not too long ago, starting with the paper (Beirão-da-Veiga et al. in SIAM J Numer Anal 51:794–812, 2013) related to elasticity in solid mechanics. The virtual element method allows to revisit the construction of different elements, however has so far not applied to space-time formulations for one-dimensional structural elements like strings, trusses and beams. Here we study several VEM elements suitable for space-time formulations that are build upon the Hamilton’s principle. It will be shown that these elements can be easily incorporated in classical finite element codes since they have the same number of unknowns. Furthermore, we show that the property of VEM to deal with non-conforming meshes is of special interest for holistic space time formulation: VEM formulations allow locally varying time discretizations (time increments) in a natural and efficient way.

This work addresses an efficient low order 3D virtual element method for elastic–plastic solids undergoing large deformations. Virtual elements were introduced in the last decade and applied to various problems in solid mechanics. The successful application of the method to non-linear problems such as finite strain elasticity and plasticity in 2D leads naturally to the question of its effectiveness and robustness in the third dimension. This work is concerned with the extensions of the virtual element method to problems of 3D finite strain plasticity. Low-order formulations for problems in three dimensions, with elements being arbitrary shaped polyhedra, are considered. The formulation is based on minimization of a pseudo energy expression, with a generalization of a stabilization techniques, introduced for two dimensional polygons, to the three-dimensional domain. The resulting discretization scheme is investigated using different numerical examples that demonstrate efficiency, accuracy and convergence properties. For comparison purposes, results of the standard finite element method are also demonstrated.

Virtual element methods (VEM) are the latest evolution of the Mimetic Finite Difference Method and can be considered to be more close to the Finite Element approach. They combine the ductility of mimetic finite differences for dealing with rather weird element geometries with the simplicity of implementation of Finite Elements. Moreover, they make it possible to construct quite easily high-order and high-regularity approximations (and in this respect they represent a significant improvement with respect to both FE and MFD methods). In the present paper we show that, on the other hand, they can also be used to construct DG-type approximations, although numerical tests should be done to compare the behavior of DG-VEM versus DG-FEM.