Nonlinear Finite Element Scheme Research Articles

Discrete strong extremum principles for finite element solutions of advection‐diffusion problems with nonlinear corrections

AbstractA nonlinear correction technique for finite element methods of advection‐diffusion problems on general triangular meshes is introduced. The classic linear finite element method is modified, and the resulting scheme satisfies discrete strong extremum principle unconditionally, which means that it is unnecessary to impose the well‐known restrictions on diffusion coefficients and geometry of mesh‐cell (e.g., “acute angle” condition), and we need not to perform upwind treatment on the advection term separately. Moreover, numerical example shows that when a discrete scheme does not satisfy the strong extremum principle, even if it maintains the global physical bound, non‐physical numerical oscillations may still occur within local regions where no numerical result is beyond the physical bound. Thus, it is worth to point out that our new nonlinear finite element scheme can avoid non‐physical oscillations around sharp layers in advection‐dominate regions, due to maintaining discrete strong extremum principle. Convergence rates are verified by numerical tests for both diffusion‐dominate and advection‐dominate problems.

Fast algorithm based on TT-M FE method for modified Cahn–Hilliard equation

We consider numerical methods for solving the modified Cahn–Hilliard equation involving strong nonlinearities. A fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme is proposed to overcome the time-consuming computation of the nonlinear terms. The TT-M FE algorithm includes three main steps: Firstly, a nonlinear FE scheme is solved on a coarse time-mesh τc. Here, the FE method is used for spatial discretization and the implicit second-order 𝜃 scheme (containing both implicit Crank–Nicolson scheme and second-order backward difference method) is used for temporal discretization. Secondly, the Lagrange’s interpolation is used to obtain the interpolation result on the fine time-mesh. Finally, a linearized FE system is solved on a fine time-mesh τ (τ<τc). The stability analysis and priori error estimates are provided in detail. Numerical examples are given to demonstrate the validity of the proposed scheme. The TT-M FE method is compared with the traditional Galerkin FE method and it is evident that the TT-M FE method can save the calculation time.

A Fast Time Two-Mesh Algorithm for Allen–Cahn Equation

In this article, we consider numerical methods for solving Allen–Cahn equation involving strong nonlinearities. A fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme is proposed to overcome the time-consuming computation of nonlinear terms. The TT-M FE algorithm includes the following three main steps: Firstly, a nonlinear FE scheme is solved on a coarse time mesh $$\tau _{c}$$. Here, the FE method is used for spatial discretization and the implicit second-order backward difference scheme is used for temporal discretization. Secondly, the Lagrange’s interpolation is used to obtain the interpolation result on the fine grid. Finally, a linearized FE system is solved on a fine time mesh $$\tau (\tau < \tau _{c})$$. The stability analysis and priori error estimates are provided in detail. Numerical examples are given to demonstrate the validity of the proposed scheme. The TT-M FE method is compared with the traditional Galerkin FE method, and it is evident that the TT-M FE method can save the calculation time.

Influence of the 3D material tailoring on snap-through and snap-back post-buckling behaviors of steel-wire-reinforced hybrid 3D graded orthotropic shallow cylindrical panels

Simultaneous application of composite layers and steel bars/wires may considerably increase buckling strength of the civil structures. In the present paper, influence of the material heterogeneity on snap-through and snap-back post-buckling behaviors of the steel-wire-reinforced hybrid functionally graded orthotropic shallow cylindrical panels is investigated. In this regard, various arrangements are adopted for the steel-wires to constitute 3D (simultaneous radial, circumferential, and axial) gradations of the materials within some layers in addition to using some traditional composite layers in between. The non-linear governing equations of post-buckling are derived based on the principle of minimum potential energy. Sanders’ shell theory is utilized to ensure that the results are reliable in the deep post-buckling region. A nonlinear finite element scheme is employed to trace the post-buckling paths using the modified arc-length method. Finally, some parametric studies in terms of material arrangement, size of the embedded steel wires, the overall lamination scheme, and geometric parameters of the shell are performed. Results emphasize the significant effects of the steel wires on the post-buckling strength and indicate the advantage of using steel fibers with exponential or uniform longitudinal distributions and linear transverse distribution, for weight optimization.

Wetting and Drying of Concrete: Modelling and Finite Element Formulation for Stable Convergence

The simulation of moisture movement in concrete is vital for estimating its durability performance under the given conditions of service. A robust analysis of the nonlinear phenomenon relies on the implementation of an efficient numerical algorithm. This entails a comprehension of the underlying mathematical models, the numerical analysis scheme and the related issues of stability and convergence. This paper describes a computational scheme to facilitate the estimation of moisture distribution in concrete subjected to wetting–drying exposure due to intermittent rains, an exposure type which severely impairs the durability performance of structural concrete under tropical climatic conditions. Beginning with a brief review of the conventional moisture transport models for concrete, this paper proposes a modified model using dimensionless parameters, which enhances the computational efficiency of the delineated scheme. This paper also describes the formulation of a one-dimensional, nonlinear finite element scheme for the modified model and discusses the issue of numerical oscillations concomitant to the numerical analysis of the problem. The suitability of the first-order element in providing stable convergence when implemented in conjunction with a lumped-mass scheme has been mathematically interpreted. The validity of the delineated proposition has been illustrated using a test problem.

Continuum modeling of charging process and piezoelectricity of ferroelectrets

Ferroelectrets in the form of electrically charged micro-porous foams exhibit a very large longitudinal piezoelectric coefficient d33. The structure has hence received wide application interests as sensors particularly in acoustic devices. During charging process, electrical breakdown (Paschen breakdown) takes place in the air pores of the foam and introduces free charge pairs. These charges are separated by electrostatic forces and relocated at the interfaces between the polymer and the electrically broken-down medium, where they are trapped quasistatically. The development of this trapped charge density along the interfaces is key for enabling the piezoelectricity of ferroelectrets. In this article, an internal variable based continuum model is proposed to calculate the charge density development at the interfaces, whereas a Maxwell stress based electromechanical model is used for the bulk behavior, i.e., of the polymer and of the medium where the Paschen breakdown takes place. In the modeling, the electrostatic forces between the separated charge pairs are included, as well as the influence of deformation of the solid layers. The material models are implemented in a nonlinear finite element scheme, which allows a detailed analysis of different geometries. A ferroelectret unit with porous expanded polytetrafluoroethylene (ePTFE) surrounded by fluorinated ethylene propylene is studied first. The simulated hysteresis curves of charge density at the surfaces and the calculated longitudinal piezoelectric constant are in good agreement with experimental results. Simulations show a strong dependency of the interface charge development and thus the remnant charges on the thicknesses of the layers and the permittivity of the materials. According to the calculated relation between d33 and the Young's modulus of ePTFE, the value of the Young's modulus of ePTFE is identified to be around 0.75 MPa, which lies well in the predicted range of 0.45 to 0.80 MPa, determined from the dielectric resonance spectra in the work of Zhang et al. [X. Q. Zhang et al., J. Appl. Phys. 108, 064113 (2010)]. To show the potential of the models, it is also applied to simulation of ferroelectrets with a lens shape. The results indicate that the electrical breakdown happens in a sequential manner, and the local piezoelectric coefficient varies with position. Thereby, the middle point on the surface exhibits the maximum d33. The simulation results obtained by the proposed models will provide insight for device optimization.

Elastohydrodynamic lubrication of piston rings

A non-linear finite-element scheme, based on the Newton-Raphson-Murty algorithm, was used to analyze the problems of piston rings according to theories of both elastohydrodynamic and hydrodynamic lubrication. The maximum film pressure and the minimum film thickness were obtained and depicted as functions of the crank angle. For hydrodynamic lubrication the results obtained were compared with those found using the method suggested by Dowson and Ruddy; there was good consistency between them. In this way the accuracy and reliability of the proposed method are demonstrated. Furthermore, the results evaluated using the theory of elastohydrodynamic lubrication give a larger minimum oil film thickness and a flatter pressure distribution than those evaluated using the theory of hydrodynamic lubrication.

Failure Strength of Nonlinearly Elastic Composite Laminates Containing a Pin Loaded Hole

A method is presented for calculating the failure strengths and failure modes of composite laminates containing a pin loaded hole for materials exhibiting nonlinearly elastic behavior. The analysis follows that proposed by Chang, Scott, and Springer for linearly elastic composites with two notable exceptions: a) the shear stress-shear strain relation in each ply is taken to be nonlinear with the form proposed by Hahn and Tsai, and b) the Yamada-Sun failure criterion is modified to include nonlinear effects. Numerical results, generated using a nonlinear finite element scheme, were compared to data. These comparisons show that for laminates exhibiting nonlinear behavior the present analysis provides the failure strengths and failure modes more accurately than the previous method employing a linear stress-strain analysis.