Weak Elements Research Articles

Optimal order uniform convergence of weak Galerkin finite element method on Bakhvalov-type meshes for singularly perturbed convection dominated problems

In this paper, we propose a weak Galerkin finite element method (WG-FEM) for solving two-point boundary value problems of convection-dominated type on a Bakhvalov-type mesh. A special interpolation operator which has a simple representation and can be easily extended to higher dimensions is introduced for convection-dominated problems. A robust optimal order of uniform convergence has been proved in the energy norm with this special interpolation using piecewise polynomials of degree $k\geq 1$ on interior of the elements and piecewise constant on the boundary of each element. The proposed finite element scheme is {parameter-free formulation} and since the interior degrees of freedom can be eliminated efficiently from the resulting discrete system, the number of unknowns of the proposed method is comparable with the standard finite element methods. An optimal order of uniform convergence is derived on Bakhvalov-type mesh. Finally, numerical experiments are given to support the theoretical findings and to show the efficiency of the proposed method.

A modified weak Galerkin finite element method for parabolic equations on anisotropic meshes

In this paper, we study a modified weak Galerkin finite element method (MWG-FEM) on anisotropic triangular meshes for a class of parabolic equations. Different from conventional weak Galerkin methods, the MWG-FEM replaces the boundary functions by the average of interior functions, which possesses flexibility in the approximation functions and mesh generation. Moreover, MWG-FEM is compatible with anisotropic meshes and suitable for solving problems with anisotropic property. By using the anisotropic interpolation and projection operators, the optimal order error estimate in L2 norm is derived. Numerical experiments are performed to demonstrate the stability and efficiency of the method.

Coming to terms with DORA

The European Union's Digital Operational Resilience Act (DORA) tackles what is currently seen as a weak element in the financial sector – third-party tech suppliers. But does it affect you? And if so, how should you respond?

The Impact of Climate Change and Window Parameters on Energy Demand and CO2 Emissions in a Building with Various Heat Sources

This article presents an original study on the impact of climate change and the area of windows Awi (factor X1), the thermal transmittance coefficient of windows Uwi (factor X2), and the coefficient of total solar transmittance factor of the glazing ggl (factor X3) on the index of annual usable energy demand for heating EUH (function Y) of a single-family residential building in the climatic conditions of Bialystok (Poland), which were loaded with an equal gradual increase in average monthly external temperature by Δθe,n (factor X4). Based on the results of the computational experiment, a deterministic mathematical model of this dependence was developed, and the effects of selected factors on the Y function were analyzed for the considered climatic conditions. Moreover, in cases of selected variants, the influence of the energy source on the amount of final energy used and CO2 emissions was studied. It was found that an increase in the average monthly external temperature reduces the EUH of the tested building by 8.4% per every 1 °C of increase in Δθe,n. The reduction in CO2 emissions as a result of climate change is visible for systems with low efficiency and high emission factors (wood boiler), while in the case of pro-ecological high-efficiency systems (with a ground-source pump heat) it is inappreciable. Due to the need to decarbonize buildings, knowledge about the impact of the properties of windows, which are the weakest element in terms of heat loss through the building envelope, as well as the type of heat source on heat demand and CO2 emissions, is very important for engineers and designers when making the correct decisions.

Weak form quadrature element analysis of spatial geometrically exact composite beams with torsional warping

Spatial thin-walled laminated composite beams under large deformations with torsional warping effects are studied. A novel two-dimensional cross-section model for composite beams with shear and torsion-warping deformations is proposed wherein the three-dimensional warping and coupling effects caused by inhomogeneous anisotropic materials are considered. A weak form quadrature element formulation is established for nonlinear analysis of geometrically exact composite beams with torsional warping and shear deformations. By combining the two-dimensional cross-section model and the nonlinear one-dimensional beam model, the three-dimensional stress and strain fields can be obtained. Numerical examples are provided to investigate the torsional warping effects and verify the effectiveness of the present formulation. Results of the present beam formulation which are achieved at a significantly lower computational cost are in excellent agreement with those of shell and solid models of commercial codes.

Elastic buckling analysis of side-cracked thin plates by weak form quadrature elements

A weak form quadrature element formulation is established for elastic buckling analysis of thin plates with side cracks. The entire problem domain is divided into a square subdomain around the crack tip and several quadrilateral subdomains elsewhere. The stress distribution of a cracked plate is determined in pre-buckling analysis. In either pre-buckling analysis or buckling analysis of a cracked plate, the potential energy of the inner square subdomain is computed using asymptotic displacement expansion around the crack tip, while the potential energy elsewhere is computed using quadrilateral weak form quadrature elements. A generalized eigenvalue problem is dealt with to determine the critical loads and buckling modes. The first ten non-dimensional buckling loads and critical buckling modes for several typical problems are given to validate the present formulation.

ARTIFICIAL INTELLIGENCE AND ITS ROLE IN INTERNATIONAL MANAGEMENT

Artificial Intelligence (AI) represents the ability that technologies or machines have to copy human intelligence as close as possible in order to solve problems and achieve goals. Artificial intelligence systems adapt, analyze data, observe future actions based on existing information and operate autonomously. An interesting change has occurred over time. In the past, the focus was on the hardware, while the software was considered a weak element. Over time, the software element developed, and over time hardware engineers adapted to the evolution becoming software engineers. Algorithms are used to make predictions in almost any field, and if used correctly, the predictions and results are beneficial and commendable. The take-up of these artificial intelligence applications in public institutions is useful to all. Therefore, developing and perfecting basic human skills is important in the long run. Above all, technology enables work to become more human. For managers, leaders or directors it has a tremendous result. It should be pointed out that starting from the first light bulb up to the emergence of the smartphone, technology has evolved. The element that never changes is the people behind the technology, while the most important aspect is that artificial intelligence is changing the working world.

Recessed Reinforced Concrete Structures to Strengthen Weak Foundations

Ensuring reliable operation of high-speed lines is an important geotechnical task. To ensure normalized values of settlement of the roadway in the presence of weak engineering-geological elements in the bases, the issues of increasing their bearing capacity are especially relevant. To do this, modern construction has a number of geotechnical technologies, as a rule, these include buried reinforced concrete structures immersed in the ground. ready-made or manufactured in the ground.

Weak Galerkin finite element methods with and without stabilizers for H(div;Ω)${\bf H}(\mbox{div}; \Omega )$‐elliptic problems

AbstractIn this article, we propose the weak Galerkin (WG) finite element schemes for ‐elliptic problems with and without stabilizers. Optimal orders of convergence are established for the WG approximations in both discrete energy norm and L2 norm. Removing stabilizers from WG finite element methods will simplify the formulations, reduce programming complexity, and may also speed up the computation time. More precisely, for sufficiently smooth solutions, we have proved the supercloseness of order two for the stabilizer free weak Galerkin finite element solution. Several numerical tests are presented to demonstrate the effectiveness of our method.

Semi-Discrete and Fully Discrete Weak Galerkin Finite Element Methods for a Quasistatic Maxwell Viscoelastic Model

Semi-Discrete and Fully Discrete Weak Galerkin Finite Element Methods for a Quasistatic Maxwell Viscoelastic Model

Torsional warping analysis of arbitrary beam cross-sections by weak form quadrature elements

Torsional warping is essential for modern beam theories. The commonly used finite element method generally suffers from low convergence rates. A weak form quadrature element formulation for torsional warping analysis of beams with arbitrary cross-section is established. Two numerical examples are given to verify the effectiveness of the present formulation. Results demonstrate that the present formulation is of high accuracy and high efficiency.

L2 estimates for weak Galerkin finite element methods for second-order wave equations with polygonal meshes

In this paper, we describe weak Galerkin finite element methods for solving hyperbolic problems on polygonal meshes. We propose both semidiscrete and fully discrete schemes to numerically solve the second-order linear wave equation. For the time discretization, we have used implicit second order Newmark scheme. For sufficiently smooth solutions, optimal order error estimate in the L2 norm is shown to hold as O(hk+1+τ2), where h is the mesh size and τ the time step. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the proposed method.

Generalized weak Galerkin finite element methods for biharmonic equations

The generalized weak Galerkin (gWG) finite element method is proposed and analyzed for the biharmonic equation. A new generalized discrete weak second order partial derivative is introduced in the gWG scheme to allow arbitrary combinations of piecewise polynomial functions defined in the interior and on the boundary of general polygonal or polyhedral elements. The error estimates are established for the numerical approximation in a discrete H2 norm and a L2 norm. The numerical results are reported to demonstrate the accuracy and flexibility of our proposed gWG method for the biharmonic equation.

A weak form quadrature element formulation of geometrically exact strain gradient shells

A geometrically exact strain gradient shell model and its weak form quadrature element formulation are proposed. The governing equations, boundary conditions and corresponding variational form of the shell model are presented. The generalized differential quadrature analogue is employed for derivative approximations of displacements and rotations with C1 continuity conditions brought about by the second-order strain gradient terms. A novel scheme based on auxiliary vectors is developed to represent and update the derivatives of constrained rotations on element edges for the numerical approach. The present quadrature shell element is feasible to incorporate strain gradient terms and free from shear and membrane locking problems due to its adjustable high-order approximation property in nonlinear shell analysis. To demonstrate the potential of this formulation for nonlinear analysis of shells with size effects, five benchmark numerical examples are presented. The results verify the feasibility of this formulation and illustrate the strain gradient effects on elastic shells undergoing large displacements and rotations.

A simplified primal‐dual weak Galerkin finite element method for Fokker–Planck‐type equations

AbstractA simplified primal‐dual weak Galerkin (S‐PDWG) finite element method is designed for the Fokker–Planck‐type equation with nonsmooth diffusion tensor and drift vector. The discrete system resulting from S‐PDWG method has significantly fewer degrees of freedom compared with the one resulting from the PDWG method proposed by Wang‐Wang. Furthermore, the condition number of the S‐PDWG method is smaller than the PDWG method (Wang‐Wang) due to the introduction of a new stabilizer, which provides a potential to design fast algorithms. Optimal‐order error estimates for the S‐PDWG approximation are established in the norm. A series of numerical results are demonstrated to validate the accuracy and efficiency of the S‐PDWG method.

Short-term analysis of extreme wave-induced forces on the connections of a floating breakwater

In the design of floating breakwaters, assessment of the extreme wave-induced loads on the connections, their weakest element, is required to ensure structure survival. This case study provides guidelines for estimating the extreme wave-induced forces on the connections of a floating breakwater. A Boundary Element Method (BEM) solver was applied to obtain the time-domain response of an array of five pontoons anchored to the sea bottom with elastic mooring lines. The hydrodynamic behaviour of the structure was assessed for short-duration sea states with different wave peak periods and oblique wave directions. Two peak selection criteria were applied to obtain force distributions, and several different probability density functions (PDF) were fitted to the resulting data. The extreme wave-induced forces on every connection and sea state were estimated for two different exceedance levels during a typical 3-h sea state. Based on the results, combination of the Peaks Over Threshold (POT) method and generalized Pareto distribution results is proposed for estimating the wave-induced design forces on the connections of floating pontoon breakwaters.

Weak Galerkin finite element method for linear poroelasticity problems

In this paper, we develop a weak Galerkin (WG) finite element method for a linear poroelasticity model where weak divergence and weak gradient operators defined over discontinuous functions are introduced. We establish both the continuous and discrete time WG schemes, and obtain their optimal convergence order estimates in a discrete H1 norm for the displacement and in H1 and L2 norms for the pressure. Finally, we present some numerical experiments on different kinds of meshes to illustrate the theoretical error estimates, and furthermore verify the locking-free property of our proposed method.

Research of orthotropic composites failure taking into account their structural stochasticity

The object of the study is the construction of the reliability assessing algorithm for the orthotropic composite plate, taking into account the stochasticity of its structure under the conditions of plane deformation. The plate consists of a matrix and reinforcement elements. The main orthotropic directions of the material coincide with the directions of the loading. The conducted studies are based on the failure criteria expressed through the components of macro stresses. The hypothesis of the weakest link is used, which for the case of the statistical theory of strength sounds like this: the ultimate (failure) loading for an orthotropic composite plate is equal to the ultimate loading for its weakest element. Defects-cracks are characterized by independent random variables – the half-length and the orientation angle between the defect line and the axis of orthotropy with a higher modulus of elasticity. The proposed model of orthotropic composite material corresponds to known experimental studies epoxy phenolic fiberglass on the cord glass fiber. The distribution probabilities density of defect orientation takes into account their predominant orientation in the direction of reinforcement. On the basis of the obtained composite failure loading integral probability distribution function, the construction and study of the dependence of the plate failure probability diagrams for different number of cracks, structural inhomogeneity and type of applied loading was carried out. Complex application of the composite materials fracture mechanics deterministic solution and the methods of probability theory and mathematical statistics allows for a more adequate assessment of the composite materials reliability, taking into account the stochasticity of their structure. The main content of this work is the construction and analysis of dependence of stochastically defective reinforced composite materials failure probability diagrams. The obtained results make it possible to evaluate the reliability of orthotropic stochastically defective materials under conditions of plane deformation. The algorithm of a compatible combination of defectiveness and randomness of the orthotropic composite material structure makes it possible to qualitatively investigate its failure under various types of applied loading.

Convergence and optimality of an adaptive modified weak Galerkin finite element method

AbstractAn adaptive modified weak Galerkin method (AmWG) for an elliptic problem is studied in this article, in addition to its convergence and optimality. The modified weak Galerkin bilinear form is simplified without the need of the skeletal variable, and the approximation space is chosen as the discontinuous polynomial space as in the discontinuous Galerkin method. Upon a reliable residual‐based a posteriori error estimator, an adaptive algorithm is proposed together with its convergence and quasi‐optimality proved for the lowest order case. The primary tool is to bridge the connection between the modified weak Galerkin method and the Crouzeix–Raviart nonconforming finite element. Unlike the traditional convergence analysis for methods with a discontinuous polynomial approximation space, the convergence of AmWG is penalty parameter free. Numerical results are presented to support the theoretical results.

Study on failure pattern evaluation method of transmission tower under wind load

To investigate the association between the failure pattern and the proportion of external load in the tower segment, the capacity experiment of the scaled segment model is designed in this paper, and the FE model of the experiment of the segment is build and compared with experiment results. The pattern of the tower segment failed is evaluated by the relationship of PH(horizontal load) and MP (bending moment). The critical points of P-MP curve were analyzed by selecting the capacity of diagonal and main elements. And the failure assessment theory for the tower segment was established. The results indicate that the FE model of the experiment segment can reflect the experiment loading process effectively. The capacity of the diagonal element has a significant impact on the tower’s PH, and the main element’s bearing capacity is an important factor affecting the flexural capacity. Using the proposed method, the weak positions and elements of the structure can be judged.