Bilinear Elements Research Articles

Superconvergence analysis of an energy-conserving Galerkin FEM for the two-dimensional sine-Gordon equation

In this paper, the superconvergence error analysis of an energy-conserving Galerkin fully discrete scheme is proposed and investigated for the two-dimensional sine-Gordon equation. The unique solvability of the numerical scheme as well as the energy conservation are studied firstly. Then, based on the special property of the bilinear element on the rectangular mesh and the superclose estimate between interpolation and Ritz projection of the exact solution in H 1 -norm, the global superconvergence result in H 1 -norm is obtained in terms of a post-processing technique. Finally, numerical results are provided to confirm the energy conservation and superconvergence properties.

A new linearized second-order energy-stable finite element scheme for the nonlinear Benjamin-Bona-Mahony-Burgers equation

In this paper, a novel linearized second-order energy-stable fully discrete scheme for the nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equation is introduced, along with a superconvergence analysis of the conforming finite element method (FEM). Through skillful decomposition of the nonlinear term u∇⋅u, a new linearized second-order fully discrete scheme is developed. This scheme requires fewer iterations and exhibits higher computational efficiency compared to the nonlinear approach. Furthermore, it maintains energy stability, enabling direct numerical solution boundedness in the H1-norm, which represents an improvement over the L∞-norm boundedness reported in previous literature. Secondly, relying on this boundedness and the high-precision results of the bilinear element, we derive the unconditional error estimates for superclose and global superconvergence without any restrictions on the time step size Δt and spatial mesh size h. Finally, numerical examples are presented to validate the theoretical analysis.

Robust convergence result of discontinuous Galerkin stabilization method for two-dimensional reaction–diffusion equation with discontinuous source term

A reaction–diffusion problem with discontinuous source term and Dirichlet's boundary conditions on the unit square is considered in this paper. The proposed problem has been discretized using a combination of standard Galerkin finite element method (FEM) and non-symmetric discontinuous Galerkin finite element method with an interior penalty (NIPG) with bilinear elements. Layer adapted mesh of Shishkin type has been utilized to discretize the domain. Standard Galerkin FEM is applied on the layer part of the domain where the domain is dense enough and NIPG is applied to the outside layer part. By means of special choice of discontinuity-penalization parameters, the scheme is proved to be uniformly convergent of order O ( ε 1 / 4 N − 1 + N − 1 ln ⁡ N ) . Numerical tests are carried out in support of theoretical findings.

Superconvergence analysis of an [formula omitted]-Galerkin mixed FEM for Klein–Gordon–Zakharov equations with power law nonlinearity

This paper aims to study an H1-Galerkin mixed finite element method (FEM) for Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity. The bilinear element and zero-order Raviart–Thomas element (Q11/Q10×Q01) are taken as approximation spaces for the original variables p, ϕ and the auxiliary variables u=∇p and ψ=∇ϕ, respectively. The supercloseness and superconvergence estimates of order O(h2+τ2) are presented for p and ϕ in the H1 norm and u and ψ in the L2 norm, which improve the known results that the original variable ϕ can only reach optimal error estimate shown in numerical experiments without theoretical analysis. Here h is the subdivision parameter and τ is the time step. Finally, some numerical results are provided to support the theoretical analysis.

Superconvergence error analysis of Ciarlet-Raviart mixed finite element method for damped Boussinesq equation

The primary objective of this paper is to investigate the error analysis for the damped Boussinesq equation with the Ciarlet-Raviart mixed finite element method, focusing on achieving superconvergence. By employing a skillful mean technique and a high precision estimate of the bilinear element, as well as combining with projection and interpolation technique, we first obtain a superclose error estimate between interpolation of the exact solution and numerical solution. Subsequently, by applying an interpolation post-processing approach, we efficiently derive a global superconvergence result. Finally, several numerical results are presented to support our theoretical analysis.

Nonconforming mixed finite element method for the diffuse interface model of two-phase fluid flows

In this work, we consider a nonconforming mixed finite element method (MFEM) for a parabolic system consisting of the Cahn-Hilliard equations and the Navier-Stokes equations, which describes a diffuse interface model for the flow of two incompressible and immiscible fluids. We focus on the superconvergence analysis for the corresponding first order convex splitting scheme, where the bilinear element is used to approximate the Cahn-Hilliard equation in space, and the constrained nonconforming rotated (CNR) Q1 element together with the Q0 constant element to the Navier-Stokes equations. By use of the high accuracy properties of the elements and some specific techniques, the superclose results of order O(h2+τ) for the concentration ϕ and the chemical potential ω in the H1-norm, the velocity u in the broken H1-norm and the pressure p in L2-norm are strictly proved. Here, h is the subdivision parameter and τ, the time step. It is worth to say that we derive a new high accuracy character through the integral identity technique for the chosen elements, which plays a quite significant role in getting the final error results. Furthermore, the global superconvergence results of the corresponding variables are concluded by the interpolated postprocessing technique. Numerical tests are provided to justify the performance of the theoretical analysis.

Convergence and superconvergence analysis of energy-preserving Crank–Nicolson Galerkin method for the Benjamin–Bona–Mahony equation

In this paper, the convergence and superconvergence error estimates are investigated for Benjamin–Bona–Mahony equation with energy-preserving Crank–Nicolson Galerkin scheme. Firstly, a priori bound of the numerical solution is obtained due to the energy conservation property of the numerical scheme. Secondly, the existence and uniqueness of the numerical solution is proved rigorously. Then, the superclose error estimate between the numerical solution and Ritz projection of the exact solution in -norm is derived by skilfully dealing with the nonlinear term. Moreover, the superclose error estimate between numerical solution and interpolation of the exact solution in -norm is acquired with a novel approach, which is the combination of the interpolation operator and Ritz projection operator as well as high accuracy error estimate of the bilinear element. Furthermore, the superconvergence error estimate in -norm is obtained with a simple and efficient interpolation post-processing approach. Finally, some numerical results are provided to verify the theoretical analysis.

COMPUTATION OF QUASISTATIC ELECTROMAGNETIC FIELDS OF COMPLEX ELECTRIC POWER FACILITIES

: Accurate computation of low-frequency quasistatic electromagnetic fields is important in the design of electricity transmission and conversion devices. This work presents the theoretical groundwork of a computation using integral equations. The procedures for solving these equations are defined by applying the method of moments and the point matching method. The unknown distribution of the source function density on the thin-wire elements is approximated by the third degree polynomials. On rectangular planes the approximation of an unknown source function is achieved by means of bilinear elements. A program has been developed for the parallel electromagnetic field computation based on the described method.

Unconditional superconvergence analysis of an energy-stable finite element scheme for nonlinear Benjamin–Bona–Mahony–Burgers equation

In this paper, an energy-stable Crank–Nicolson fully discrete finite element scheme is proposed for the Benjamin–Bona–Mahony–Burgers equation. Firstly, the stability of energy is proved, which leads to the boundedness of the finite element solution in H^{1}-norm. Secondly, combining with the above boundedness and the special property of bilinear element, the unconditional superclose and superconvergence results are derived. Finally, numerical examples are provided to illustrate the validity and efficiency of our theoretical analysis and method.

Superconvergence analysis for nonlinear viscoelastic wave equation with strong damping

ABSTRACT In this research, with help of the bilinear finite element, superconvergence analysis is proposed for nonlinear viscoelastic wave equation with strong damping. Firstly, a temporal discrete approximate scheme is established, and assisted by the method of first hypothesis and then proof, the temporal errors are given. Secondly, a new linearized second-order fully discrete scheme is presented, by using the technique of Interpolation and Ritz-prejection combination in the error estimations and analysis process, rigorous proofs are provided for the unconditional superclose estimates in -norm with order , here h and τ denote mesh size and time step, respectively. Finally, recurring to numerical examples, the correctness of the theoretical analysis is further demonstrated.

Curved node-to-face contact schemes for higher-order finite elements in lumped-mass explicit methods

Contact is crucial in many lumped-mass explicit analyses, such as for high-rate impact, and enforcement methods for higher-order elements are generally less mature than for first-order ones. In this paper, basic lumped-mass explicit capabilities are pursued for three-dimensional contact/impact that account for arbitrary-order elemental curvature but with reasonable computational costs. Deformable-to-rigid and deformable-to-deformable surface-to-surface contact are considered with both triangular and/or quadrilateral faces for large deformation/sliding. Double-pass node-to-face contact searching and enforcement approaches are presented for general three-dimensional C0 higher-order Lagrange type elements in nonlinear solid dynamics using typical lumped-mass central difference time integration. This study uses classical kinematically compliant and penalty methods as well as presents modified versions better suited for higher-order Lagrange type elements. It also presents a general outer-loop iterative framework for increased accuracy and examines the simplifying practice of partitioning (subdivision) of higher-order element faces into first-order ones. Assessments are made with benchmark and practical high-rate impact applications involving elastic, hyperelastic, and elastic–plastic material models. Although subdivision into bilinear elements can sometimes be effective, it is demonstrated that they can also fail and that more precise treatment of even only modest amounts of curvature can be more reliable and improve performance.

Numerical Analysis of Structural Performance of Concrete-GFRP Composite I-Beam

The concrete-GFRP composite beams have received extensive attention in civil engineering. However, the ambiguity of the fracture, debonding of the interface, and the GFRP profile limit the precise design of the composite beam. This article presents a comprehensive numerical study for the structural performance of composite pultruded GFRP beams to provide a better understanding of the mechanism of interfacial debonding and GFRP matrix fracture. The failure and delamination process of pultruded GFRP for anisotropy of materials is modeled using the Hashin criteria. The bond–slip behavior between the concrete slab and the top flange of the GFRP I-beam is simulated by the bilinear cohesive interface element. The availability and accuracy of the finite element model are verified by comparison with the four-point bending test results of the pure GFRP I-beam and composite beams as well. Based on the proposed comprehensive finite element model, the effects of the strength, thickness, and width of the concrete slab and the shear-span ratio of the beam on the structural behavior of the composite beam are studied. According to the parametric analysis, the excessive high strength of concrete, the width, and/or thickness of the concrete slab would lead to shear failure of the slab rather than significantly increasing the ultimate load of the composite beam. When having a small shear-span ratio, the matrix fracture and delamination will occur in the web of the GFRP profile. In addition, the height of the I-profile web has a significant effect on the stress and strain distribution of the composite beam. These parametric analyses could provide the numerical basis for the design of the GFRP composite beams.

2D acoustic wavefield simulation by improved stiffness and mass matrices of bilinear square element

Abstract In the frequency domain, we optimize the entries of the element stiffness matrix computed on a square element with the purpose of approximating the 2D acoustic wave equation with the second spatial order. The optimized matrices are computed from the minimization of the normalized phase velocity as a function of propagation angle and the number of grid points per wavelength, leading to a remarkable reduction in the numerical error. The optimized number of 4.1 sample points in a wavelength, with a maximum error in velocities <0.3%, offering the simulation of large-scale realistic models. The superior performance of the optimized matrices to those from the lumped, consistent and eclectic matrices, is evident from the numerical examples presented, with consequent reduction not only in the numerical dispersion/anisotropy but also in an improved resolution. It is noteworthy that optimized matrices give modeling results with better accuracy than models from the high-order spectral element method. Our methodology is easily extendable to the accurate discretizing of the general 3D elastic wave equation that optimizes the bandwidth of the impedance matrix, availing computational resources for accurately modeling the compressional and shear wave velocities, density, as well as multi-parameter inversion.

Large Amplitude Axisymmetric Free Vibration of a Closed Two-Dimensional Spherical Pressure Vessel with a Constrained Volume

The nonlinear axisymmetric vibrational modes of a closed two-dimensional spherical pressure vessel with a constrained volume are presented using the fundamental forms of surfaces. The strain energy of the spherical pressure vessel is derived as a quadratic function of the initial Eulerian, added, and total Lagrangian strains. Two-dimensional bilinear elements described in spherical polar coordinates with the symmetric boundary condition along both sides of the meridian line are used for investigating the nonlinear axisymmetric mode shapes of a spherical pressure vessel. Numerical results were obtained by a nonlinear finite element approach and verified in the case of an empty spherical shell for both lower and upper branches. This study showed that the spherical pressure vessel gave a higher nonlinear axisymmetric natural frequency than the empty spherical shell. The elastic modulus has a large effect on the nonlinear axisymmetric frequencies of both the lower and upper branches for the spherical pressure vessel. Changing the thickness and initial internal pressure significantly affects only the lower branch of the spherical pressure vessel at high mode numbers. The results also indicate that the ratios of the nonlinear natural frequencies of the lower to upper branches decrease when the vibrational modes increase.

Nonlinear material identification of heterogeneous isogeometric Kirchhoff–Love shells

This work presents a Finite Element Model Updating inverse methodology for reconstructing heterogeneous material distributions based on an efficient isogeometric shell formulation. It uses nonlinear hyperelastic material models suitable for describing incompressible material behavior as well as initially curved shells. The material distribution is discretized by bilinear elements such that the nodal values are the design variables to be identified. Independent FE analysis and material discretization, as well as flexible incorporation of experimental data, offer high robustness and control. Three elementary test cases and one application example, which exhibit large deformations and different challenges, are considered: uniaxial tension, pure bending, sheet inflation, and abdominal wall pressurization. Experiment-like results are generated from high-resolution simulations with the subsequent addition of up to 4% noise. Local optimization based on the trust-region approach is used. The results show that with a sufficient number of experimental measurements, design variables and analysis elements, the algorithm is capable to reconstruct material distributions with high precision even in the presence of large noise. The proposed formulation is very general, facilitating its extension to other material models, optimization algorithms and meshing approaches. Adapted material discretizations allow for an efficient and accurate reconstruction of material discontinuities by avoiding overfitting due to superfluous design variables. For increased computational efficiency, the analytical sensitivities and Jacobians are provided.

Superconvergence analysis of Galerkin method for semilinear parabolic integro-differential equation

In this paper, based on the bilinear element used for spatial discretization and a linearized backward Euler scheme used for temporal discretization, the superconvergence error estimate is derived for semilinear parabolic integro-differential equation without certain time-step restrictions. The key is to derive a uniform boundness of the numerical solution in energy norm under the weaker assumption compared to previous literatures for nonlinear term. Numerical results are presented to confirm the correctness of the theoretical analysis.

An adaptive phase-field model based on bilinear elements for tensile-compressive-shear fracture

An efficient adaptive phase-field method based on bilinear elements for tensile-compressive-shear fracture is developed. On the one hand, refined meshes are needed near the crack path to obtain accurate results, and the computational efficiency is very low if a prior mesh refinement is applied, especially when the crack paths are unknown. On the other hand, conventional phase-field models are not applicable to tensile-compressive-shear fracture under complex stress states at present. In this paper, an adaptive scheme is proposed to improve the computational efficiency, in which a bilinear multi-node element is adopted to avoid using high order quadrature and shape functions when new nodes are inserted in elements, and a new multi-node triangular element is outlined to expand the scope of application of the proposed adaptive method. Another important aspect of the contribution is the development of the phase-field model for tensile-compressive-shear fracture under complex stress states, in which a universal fracture criterion is embedded. To determine the optimal parameters in the adaptive scheme and demonstrate the advantages of the proposed bilinear adaptive method, a series of numerical examples are performed for sensitivity analysis. Comparison with experimental results is also conducted to validate the proposed phase-field model for tensile-compressive-shear fracture under complex stress states.

High accuracy analysis of Galerkin finite element method for Klein–Gordon–Zakharov equations

The main aim of this paper is to propose a Galerkin finite element method (FEM) for solving the Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity, and to give the error estimations of approximate solutions about the electronic fast time scale component q and the ion density deviation r. In which, the bilinear element is used for spatial discretization, and a second order difference scheme is implemented for temporal discretization. Moreover, by use of the combination of the interpolation and Ritz projection technique, and the interpolated postprocessing approach, for weaker regularity requirements of the exact solution, the superclose and global superconvergence estimations of q in H1−norm are deduced. At the same time, the superconvergence of the auxiliary variable φ (−Δφ=rt) in H1−norm and optimal error estimation of r in L2−norm are derived. Meanwhile, we also discuss the extensions of our scheme to more general finite elements. Finally, the numerical experiments are provided to confirm the validity of the theoretical analysis.

Superconvergence analysis of a mixed finite element approximation for the nonlinear fourth-order Rosenau-RLW equation

The focus of this paper is on a implicit Backward-Euler (BE) scheme with the mixed finite element method (FEM) for the two-dimentional general Rosenau-RLW equation. In which the bilinear element is used to approximate the exact solution v and the variable p=−Δv, and the zero-order nédéléc's finite element to the variable q→=ϕ∇v, respectively. An important point is that with the proposed scheme we are able to bound the numerical solution in H1-norm, so that we can efficiently solve the nonlinear term. Moreover, the stability, existence and uniqueness of approximate solution are demonstrated. Based on the combination of the interpolation and projection technique, the superconvergence estimates of order O(h2+τ) for v in H1-norm and the introducing variable q→ in L2-norm, and optimal error estimate of order O(h+τ) for introducing variable p in H1-norm is proved. Finally, numerical example is done to certify our theoretical results.

DEFORMATION TRANSVERSE SHEAR BENDING STATE OF A THIN PLATE LAYER OF AN ANISOTROPIC GEOLOGICAL MEDIUM FROM THE ACTION OF CONCENTRATED ENERGY IMPULSES

A method is proposed for study the structural stability of the deformation state of structural blocks of the earth's crust, approximated in the form of plate layers of the geological medium when transverse shear bending from the action of concentrated energy impulses. Advances here are carried out in the two directions. First, in contrast to the previous article, the physical and mechanical model of the geological medium is endowed with anisotropic properties, which makes it possible to increase the adequacy of the obtained numerical results to the specifics of the real problem. Secondly, instead of the simplest bilinear 4-node finite elements, the special spectral non-algebraic 8-node finite iso-parametric finite elements are used, the use of which significantly increases both the accuracy of calculations and their reliability in the sense of ensuring the robustness of calculations for relatively small values of the plate thickness. It should be noted that the Finite Element Method uses exclusively only algebraic finite elements (power polynomials in the h-version and orthogonal polynomials in the p-version). It is known from approximation theory that the use of spectral non-algebraic approximations improves the quality of approximations. Therefore, their introduction into the structure of finite element calculations can improve the quality of modeling in the study of the strain-stress-state (SSS) of the geological medium. A structural block (SB) is understood as a plate layer with plan dimensions exceeding the thickness by more than 10 times. The identification of hazardous zones in the rock massive due to stress concentration is complemented by the development of mechanical, mathematical and computational tools for modeling the curvature of the earth's crust during bending based on the classical theory of Kirchhoff and refined Reissner-Mindlin theory. Test calculations have shown that the accuracy of the calculation and the quality of geometric modeling of fragments of an anisotropic geological environment based on the refined 8-node spectral finite element is significantly better than for the 8-node algebraic finite