Nonconforming Element Research Articles

Unconditional optimal [formula omitted]-norm error estimate and superconvergence analysis of a linearized nonconforming finite element variable-time-step BDF2 method for the nonlinear complex Ginzburg–Landau equation

In this paper, a linearized fully discrete scheme combining the variable-time-step two-step backward differentiation formula (VSBDF2) in time and the nonconforming finite element methods (FEMs) in space is constructed and analyzed for the nonlinear complex Ginzburg–Landau equation. A novel convergence analysis approach is proposed, which shows that the H1-norm error of this scheme can reach optimal order in space and sharp second-order convergence in time. The key ingredients of our analysis are temporal–spatial error splitting approach, a new positive definiteness property of discrete orthogonal convolution (DOC) kernels with respect to complex sequences, an energy projection operator and a discrete Laplace operator Δh related to the nonconforming element space, and a relationship between Δh and classical L2 projection operator in complex space. Furthermore, through a specific treatment of the error equation and the combination technique of interpolating operator and projection operator, we prove that the proposed scheme has the global superconvergence result of order O(τ2+h2) in H1-norm for the first time, which further improves the efficiency of numerical computation. Here τ and h are the maximum time and spatial step sizes, respectively. It should be pointed out that all these error results are rigorously established under mild assumptions of the time step, and get rid of the usual grid ratio constraints between temporal and spatial step sizes in linearized scheme. At last, numerical experiments are provided to verify the theoretical analysis.

Band gap study of periodic piezoelectric micro-composite laminated plates by finite element method and its application in feedback control

Phononic crystals are materials or structures with periodic variations of elastic constants and density. Elastic waves in phononic crystals cannot propagate within a certain frequency range, called the band gap, due to the interference of their internal structure. This article investigates how fiber orientation affects the band gap properties of periodic micro-composite laminated plates. We examined the impact of five common composite laminate configurations on the system’s band gap and discovered that the antisymmetric configuration had superior high-frequency band gap performance. Based on this finding, we further explored how the antisymmetric angle influenced the system’s band gap. Since the study was conducted at a micro scale, micro structural effects were considered. Therefore, we used the modified coupled stress elastic dynamics and a four-node quadrilateral non-conforming element to discretize the micro composite laminated plate model, which had the nodal compatibility of high-order elasticity theory. Moreover, we employed feedback control to the proposed structure to dynamically adjust the band gap width according to different practical needs. This research supports the design of periodic micro-composite laminated plates and the piezoelectric feedback control system, which can control the vibration and wave propagation behaviors of micro structures using coupled piezoelectric sensors and actuators. The piezoelectric feedback control employed two control methods: direct proportional control and acceleration control. This article compared the single and multiple control methods of these two approaches to determine the optimal control strategy.

Study of the effect of macrogeometry of rubbing steel bodies at their point contact on the tribological characteristics of lubricants under boundary lubrication conditions

The effect of the shape of contacting nonconformal elements of friction units that form an initial point contact on the tribological characteristics of lubricants during friction in the boundary lubrication mode. Point contact of rubbing bodies is realized on four-ball friction machines when testing tribological characteristics of lubricants in accordance with Russian GOST 9490–75 and GOST 23.221–84 Standards.However, these tests are carried out with the friction of standard bearing balls and does not allow taking into account the impact of materials of rubbing bodies on the results of the experiment. Therefore, two alternative designs of mandrels were developed, which allow implementation of two friction schemes: «rotating ball clamped in the spindle of the machine – three roller face planes» and «rotating ball — three cylindrical rollers». The load on the friction unit for both friction schemes, as well as in parallel on the mandrel under testing, in which three balls are installed, is the same (108 N), the spindle rotation speed is equal to 1 rpm, duration of testing is 60 min. The diameter of balls used in these experiments was 12.7 mm, and diameters of rollers were 5 and 8 mm, respectively. The material of samples was ShKh-15 tool steel or its 100Cr6 analogue. The designs of mandrels and the dimensions of samples were chosen in such a way that in all cases the contact areas of the experimental samples with the wearing (upper) ball would lie on a circle with a diameter of 7.32 mm. The tests were carried out under dry friction, and in boundary lubricated conditions in a PAO-4 polyalphaolefin oil medium, as well as in the same oil medium with the additives of oleic acid and DF-11 (a solution of zinc dialkyldithiophosphate in a low-viscosity oil). It is shown that the composition of a lubricating medium has a decisive effect on the test results (i.e., on the values of the friction coefficients and on the values of the areas of wear scars), while the geometry of the rubbing bodies, as well as the maximum Hertz stresses in the tribological contact, had a little effect on the antifriction and anti-wear characteristics of oils.

The Nation-State and the (Re)Construction of Religious, Ethnic and Gender Relations

Nationalism, nationhood, and ethnicity, as Eric Hobsbawm argued, are social processes constructed essentially from above, yet cannot be understood unless also analyzed from below.1 Inspired by European Orientalism, the intellectual advocates of Western-oriented nationalism attempted to establish a new Iranian identity based on Persian language and Iran's pre-Islamic past.2 This made Iranian nationalism an attractive ideology for some political elites, and was later endorsed by the Pahlavi dynasty (1925–79) when nationalist ideology replaced Iranianness with Persianity. In this conception, there was no room for the ethnic diversity of a nation-state that is the heir of an ancient empire. The elites’ aspirations to uniformity put pressure on members of ethnic groups to conform their way of life to a new model of Iranianness, Persianized and pro-Western. Every nonconforming element was regarded as a sign of backwardness and possible threat to the modern nation and its territorial integrity.

Comprehensive analysis of bio-inspired laminated composites plates using a quasi-3D theory and higher order FE models

A comprehensive study is carried out by employing various finite element models (FEMs) for the bending, buckling stability and free vibration analyses of bio-inspired helicoidal composite plates with various lamination schemes. A higher order quasi-3D kinematic plate theory is developed to include a shear deformation effect. The variational formulation of the problem is exploited to derive the equations of motion, element stiffness, geometrical stiffness, and mass matrices based on a non-conforming rectangular element. Three different finite elements models are derived based on non-conforming elements with different number of nodes and degree of freedom. The developed finite element model has been validated with those found in the open literature. The effects of boundary condition, lamination scheme, orthotropy ratio and aspect ratio on the mechanical response of the bio-inspired helicoidal composite plates are examined. Notably, for the lamination schemes investigated in this study, no shear locking phenomenon was observed in the analyses conducted using these FEMs. Dimensionless centre deflections, critical buckling loads and fundamental frequencies of bio-inspired helicoidal composite plates vary depending on the type of lamination scheme, boundary condition and aspect ratio. The new orientation schemes can replace the traditional ones to overcome the shear singularity and overcome the delamination defects.

Technical-Economic Opportunities of Mixture Optimization and Recycled Aptitude at a Pre-Casting Concrete Plant

Cost optimization is a key factor in the competitiveness of construction companies. Hence, in the case of a concrete manufacturing plant, the technical-economic optimization of the process involved in producing concrete is a determining factor in its activity being more competitive than that of other companies. Thus, the main objective of this experimentation is the optimization of the different concrete dosages used in a pre-casting plant. The cement volume, the water/cement (W/C) ratio, the influence of an accelerator additive and replacing coarse aggregate with recycled aggregate were analysed. As an application of the economic results, three different hypotheses were analysed for different mix combinations in a real concrete structure model. The results show that it is essential to monitor the actual initial moisture state of aggregates and the perfect definition of the effective water/cement ratio of the dosage (Wef/C) in manufacturing in order to obtain more homogeneous production. It is proposed to simplify the number of different mix proportions and reduce cement by 10–20% without decreasing resistance and functional capabilities. Regarding additive optimization, the results show that using low rates of accelerator additive significantly improves compressive strength. Replacing natural aggregates with recycled aggregates obtained from the concrete waste of non-conforming elements is suited to manufacturing recycled concrete using replacement rates of 50% and 100%. The cost analysis for a real industrial building applying the different dosing options studied shows that a cost reduction of up to 17.8% can be achieved.

Forward modeling of direct-current resistivity data on unstructured grids using an adaptive mimetic finite-difference method

An effective solver for the direct-current (DC) resistivity forward-modeling problem should be capable of accommodating arbitrary electrode layouts, adapting the mesh to topography and complex geologic structures, and adaptively refining the mesh to ensure high solution accuracy. We use the capability of the mimetic finite-difference method in naturally accommodating meshes with nonconforming elements to develop adaptive and parallel cell-based (CB) and vertex-based (VB) mimetic schemes for the forward modeling of DC resistivity data on unstructured meshes. A diffusion problem in mixed form and Poisson’s problem are solved in the CB and VB schemes, respectively. The mesh adaptivity involves an iterative h-refinement conducted by a regular subdivision of the marked elements together with a 2-irregularity condition. Goal-oriented residual- and gradient-based error estimators are used to mark the elements for refinement in the CB and VB schemes, respectively. To evaluate the accuracy and efficiency of the mimetic schemes, we use two benchmark models with analytical solutions and standard linear and quadratic finite-element solutions. The numerical results for apparent resistivity from both mimetic schemes accurately approximate the reference analytical and numerical values. Furthermore, on similar meshes, the VB mimetic scheme is found to be less demanding than the CB method in terms of computational resource requirement. We also develop a grid-based mesh generation technique based on the body-centered cubic lattice to generate high-quality initial meshes. Using benchmark examples, we validate this mesh generation method and demonstrate its versatility by discretizing a model of a thin dike where a standard Delaunay mesh generator fails to generate a mesh. Moreover, we display the effectiveness of the presented mimetic approach and the grid-based mesh generation technique using a realistic example with topography and parallelization over a large number of sources.

Nonconforming finite element Stokes complexes in three dimensions

Two nonconforming finite element Stokes complexes starting from the conforming Lagrange element and ending with the nonconforming P1-P0 element for the Stokes equation in three dimensions are studied. Commutative diagrams are also shown by combining nonconforming finite element Stokes complexes and interpolation operators. The lower order H(gradcurl)-nonconforming finite element only has 14 degrees of freedom, whose basis functions are explicitly given in terms of the barycentric coordinates. The H(gradcurl)-nonconforming elements are applied to solve the quad-curl problem, and the optimal convergence is derived. By the nonconforming finite element Stokes complexes, the mixed finite element methods of the quad-curl problem are decoupled into two mixed methods of the Maxwell equation and the nonconforming P1-P0 element method for the Stokes equation, based on which a fast solver is discussed. Numerical results are provided to verify the theoretical convergence rates.

P1$$ {P}_1 $$‐nonconforming quadrilateral finite element space with periodic boundary conditions: Part II. Application to the nonconforming heterogeneous multiscale method

AbstractA homogenization approach is one of effective strategies to solve multiscale elliptic problems approximately. The finite element heterogeneous multiscale method (FEHMM) which is based on the finite element makes possible to simulate such process numerically. In this paper we introduce a FEHMM scheme for multiscale elliptic problems based on nonconforming elements. In particular we use the noconforming element with the periodic boundary condition introduced in the companion paper. Theoretical analysis derives a priori error estimates in the standard Sobolev norms. Several numerical results which confirm our analysis are provided.

A New Family of Nonconforming Elements with $H$(curl)-Continuity for the 3D Quad-Curl Problem

A New Family of Nonconforming Elements with $H$(curl)-Continuity for the 3D Quad-Curl Problem

Improving the Bergan Wang Quadrilateral plate element by using the UI approach for analysis of isotropic and FGM sandwich plates

This work presents the improvement of Bergan Wang Quadrilateral (BWQ) element using the UI (unified and integrated) approach. In this approach, a new formulation of quadrilateral plate element is developed with a single independent variable just like what Bergan and Wang applied to develop a non-conforming BWQ element. The difference between the two approaches lies in the formulation of transverse displacement w, wherein the UI approach transverse displacement w is expressed as the sum of transverse bending, and shear displacements (wb and ws) and the transverse bending displacement wb becomes the single independent variable. The transverse shear displacement ws is defined as the second derivative of the transverse bending displacement wb. The rotations, curvatures, and shear strains are the first, second, and third derivatives of the bending displacement wb. The resulting non-conforming element, called UI-Q4 element, has proper rank, no shear locking problem, and passes the patch test. To examine the element convergence and accuracy, standard benchmark problems of isotropic plates, i.e. square plates, Razzaque and Morley skew plates, and circular plates, are used. The UI-Q4 element shows very good performance, which is observed by comparing it with MITC4, DSQK and BWQ elements. Moreover, the application of UI-Q4 element for analysing functionally graded material (FGM) sandwich plate analysis also performs well and delivers results which are quickly converged to the reference values.

A subspace of linear nonconforming finite element for nearly incompressible elasticity and Stokes flow

Abstract The linear nonconforming finite element, combined with constant finite element for pressure, is stable for the Stokes problem. But it does not satisfy the discrete Korn inequality. The linear conforming finite element satisfies the discrete Korn inequality, but is not stable for the Stokes problem and fails for the nearly incompressible elasticity problems. We enrich the linear conforming finite element by some nonconforming P1 bubbles, i.e., select a subspace of the linear nonconforming finite element space, so that the resulting linear nonconforming element is both stable and conforming enough to satisfy the Korn inequality, on HTC-type triangular and tetrahedral grids. Numerical tests in 2D and 3D are presented, confirming the analysis.

Active tuning of vibration for periodic piezoelectric micro systems: A non-local Mindlin plate finite element approach

The present paper is aimed to propose a multi-parameter feedback control method combined with couple stress elasticity to model piezoelectric micro plate coupled systems. The proposed methodology can be used to design controllers for tuning vibration and wave propagation properties of micro scale plates based on coupled piezoelectric sensors and actuators. Specifically, we use a three-parameter relationship that describes the voltage gain within the sensor-to-actuator circuit involving multiple dependence based on mass, damping and stiffness. Consequently, effect of these parameters can be simulated either independently or collectively so as to obtain the optimal control strategy with respect to the required vibroacoustic properties. Meanwhile, since micro plates are involved, the inherent microstructure effects must be accounted for. Hence, the modified couple stress elasto-dynamics is applied and the micro plate model is discretised with a four-node quadrilateral non-conforming element that offers nodal compatibility with high-order theories of elasticity. Based on the proposed numerical methodology, we investigated the feedback control parametrisation for a reference micro plate coupled system which presents significant microstructure effects. Our analysis allowed characterisation of the three control parameters based on their individual effects, and revealed that their combined effect cannot be predicted by considering direct superposition of their individual behaviours. Therefore, the proposed computational methodology provides a convenient solution for the choice and parametrisation of the feedback controller leading to tunable band gap properties of micro scale plate structures.

A priori and a posteriori error analysis of the lowest-order NCVEM for second-order linear indefinite elliptic problems

The nonconforming virtual element method (NCVEM) for the approximation of the weak solution to a general linear second-order non-selfadjoint indefinite elliptic PDE in a polygonal domain Omega is analyzed under reduced elliptic regularity. The main tool in the a priori error analysis is the connection between the nonconforming virtual element space and the Sobolev space H^1_0(Omega ) by a right-inverse J of the interpolation operator I_h. The stability of the discrete solution allows for the proof of existence of a unique discrete solution, of a discrete inf-sup estimate and, consequently, for optimal error estimates in the H^1 and L^2 norms. The explicit residual-based a posteriori error estimate for the NCVEM is reliable and efficient up to the oscillation terms. Numerical experiments on different types of polygonal meshes illustrate the robustness of an error estimator and support the improved convergence rate of an adaptive mesh-refinement in comparison to the uniform mesh-refinement.

Asymptotic expansions of eigenvalues by both the Crouzeix–Raviart and enriched Crouzeix–Raviart elements

Asymptotic expansions are derived for eigenvalues produced by both the Crouzeix-Raviart element and the enriched Crouzeix–Raviart element. The expansions are optimal in the sense that extrapolation eigenvalues based on them admit a fourth order convergence provided that exact eigenfunctions are smooth enough. The major challenge in establishing the expansions comes from the fact that the canonical interpolation of both nonconforming elements lacks a crucial superclose property, and the nonconformity of both elements. The main idea is to employ the relation between the lowest-order mixed Raviart–Thomas element and the two nonconforming elements, and consequently make use of the superclose property of the canonical interpolation of the lowest-order mixed Raviart–Thomas element. To overcome the difficulty caused by the nonconformity, the commuting property of the canonical interpolation operators of both nonconforming elements is further used, which turns the consistency error problem into an interpolation error problem. Then, a series of new results are obtained to show the final expansions.

Low-order nonconforming brick elements for the 3D Brinkman model

This work gives a general framework for the convergence analysis of low-order nonconforming elements for the 3D Brinkman model over cubical meshes, including the theory of uniform convergence with respect to the given parameters and the high accuracy analysis in the Darcy region. We apply our theory to three elements known in other literature and a newly constructed edge-face-based element to show their optimal approximation ability. Numerical examples are also provided to confirm our theory.

H$$^2$$-Korn’s Inequality and the Nonconforming Elements for The Strain Gradient Elastic Model

We establish a new H\(^2\)-Korn’s inequality and its discrete analog, which greatly simplify the construction of nonconforming elements for a linear strain gradient elastic model. The Specht triangle (Specht in Int J Numer Methods Eng 28:705–715, 1988) and the NZT tetrahedron (Wang et al. in Numer Math 106:335–347, 2007) are analyzed as two typical representatives for robust nonconforming elements in the sense that the rate of convergence is independent of the small material parameter. We construct the regularized interpolation operators and the enriching operators for both elements, and prove the error estimates under minimal smoothness assumption on the solution. Numerical results for the smooth solution, and the solution with boundary layer are consistent with the corresponding theoretical prediction.

Refined numerical simulation of a concrete cut-off wall in the thick overburden of dam foundation

It is inevitable for dam construction in hydropower exploitation to face a thick overburden foundation with high permeability. The performance of the concrete cut-off wall in an overburden foundation is the key factor to determine dam safety. However, it is difficult to conduct detailed and in-depth study on the wall under complex conditions based on traditional simulation and analysis methods. In this research, the numerical simulation of a concrete cut-off wall in thick overburden are intensively studied. The deformation, stress, and damage distribution of the concrete cut-off wall are analyzed in detail. The importance and necessity of the refined numerical simulation and analysis of a concrete cut-off wall are discussed based on the aspects of scale, grid, analysis content, and method. The results indicate that it is very necessary to conduct a three-dimensional numerical analysis of the cut-off wall. A rough numerical model can be adopted to model the cut-off wall when only deformation is concerned. However, it is necessary to carefully mesh the wall when the stress state is the key research content, and it is more economical and reliable to model the wall with a nonconforming element. Moreover, the plastic damage analysis has revealed that the wall is more liable to tensile damage and the results can be used to establish a direct connection with the anti-seepage performance. In addition, some reinforcement measures have been suggested with a detailed study of the stress generation mechanism and stress direction development process in the concrete cut-off wall. This study can provide a method basis and technical support for the safety design and evaluation of a concrete cut-off wall in the thick overburden of dam foundation.

An Optimal Multilevel Method with One Smoothing Step for the Morley Element

Abstract In this paper, a class of multilevel preconditioning schemes is presented for solving the linear algebraic systems resulting from the application of Morley nonconforming element approximations to the biharmonic Dirichlet problem. Based on an appropriate space splitting of the finite element spaces associated with the refinements and the abstract Schwarz framework, we prove that the proposed multilevel methods with one smoothing step are optimal, i.e., the convergence rate is independent of the mesh sizes and mesh levels. Moreover, the computational complexity is also optimal since the smoothers are performed only once on each level in the algorithm. Numerical experiments are provided to confirm the optimality of the suggested methods.

The simulation of high compressive stress and extrusion phenomenon for concrete face slabs in CFRDs under strong seismic loads

The evaluation of high compression zones for anti-seepage face slabs is an increasingly important task in the safety evaluation of high concrete faced rock-fill dams (CFRDs). This manuscript uses the scaled boundary finite element method (SBFEM) to construct a cross-scale model of the rockfill and bedrock in a high CFRD. Nonconforming elements are applied to independently refine the mesh locally in the concrete face slab, which allows an accurate modeling of the rotating and bending phenomenon experienced by the face slab zone. Meanwhile, the non-matching node interface is introduced to connect the locally refined face slab mesh to the rockfill zone and obtain the final cross-scale model for refined analysis. The generalized plastic constitutive model and a state-dependent elasto-plastic interface constitutive model are employed to capture the complex behaviors of rockfill and soil-structure interaction (SSI). Leveraging the above approaches, static and dynamic analyses of a high CFRD are conducted to identify high compressive stress zones and simulate the extrusion damage. The results indicate that self-weight, water pressure, friction force and seismic loads contribute to rotating and bending between face slabs, and can lead to significant local stress concentration within the surface of face slab near the longitudinal joints and may potentially extrusion damage.