Linear Finite Element Method Research Articles

Three-dimensional stress analyses of complex laminated shells with a variable-kinematics continuum shell element

This paper presents a new variable-kinematics continuum shell (VKCS) element that can be used to model laminated shell structures with arbitrary geometry in a finite element (FE) setting. The novelty is in the implementation of variable-kinematics capability in a continuum shell formulation, using Carrera’s Unified Formulation (CUF). The resultant model has completely general geometric and kinematic descriptions. In the formulation, the geometrical representation is based on a numerical isoparametric map with no simplifying assumptions on the shell geometry; whereas the element displacement fields are written in terms of the Fundamental Nuclei according to CUF. In the variable-kinematics framework, the levels of hp– and p–refinements in the through-thickness and in-plane domains are free parameters that can be varied independently. By parametrically varying in-plane mesh densities and model kinematics, model settings with good trade-offs in computational cost and desired level of accuracy can be identified. In addition to the existing literature benchmarks, we include new 3D stress benchmarks for laminated shells with complex geometrical features, such as spatially varying curvatures, non-orthogonal coordinate lines and variable thicknesses. The higher-order models yield asymptotically correct three-dimensional stresses, even in regions near singularities, without requiring numerical artefacts nor stress recovery procedures. In terms of computational efficiency, the model variants utilising high p-level require fewer total degrees of freedom (dofs) compared with linear 3D finite element method (FEM) for convergence of the 3D stress field. In terms of wider applications, the compact formulation can allow for the same computer code and model mesh to be used across a wide range of analyses for complex shell structures that requires different model fidelity, with minimal inputs from the user.

Convergence and superconvergence of variational discretization for parabolic bilinear optimization problems

In this paper, we investigate a variational discretization approximation of parabolic bilinear optimal control problems with control constraints. For the state and co-state variables, triangular linear finite element and difference methods are used for space and time discretization, respectively, superconvergence in H^{1}-norm between the numerical solutions and elliptic projections are derived. Although the control variable is not discrete directly, convergence of second order in L^{2}-norm is obtained. These theoretical results are confirmed by two numerical examples.

Superconvergence analysis of FEM for 2D multi-term time fractional diffusion-wave equations with variable coefficient

The goal of this paper is to discuss high accuracy analysis of a fully-discrete scheme for 2D multi-term time fractional wave equations with variable coefficient on anisotropic meshes by approximating in space by linear triangular finite element method and in time by Crank-Nicolson scheme. The stability is firstly proved unconditionally. In the analysis of superclose properties, how to deal with the item for variable coefficient is the main difficulty. In order to do this, a new projection operator is defined and the relationship between the proposed projection operator and interpolation operator about linear triangular finite element is deduced. Consequently, the global superconvergence result is obtained by use of interpolation postprocessing technique. The numerical examples show that the proposed numerical method is highly accurate and computationally efficient.

Superconvergence of finite element method for singularly perturbed convection–diffusion equations in 1D

In this article, a singularly perturbed convection–diffusion equation is solved by a linear finite element method on a Shishkin mesh. By means of an analysis exploiting symmetries in the convective term of the bilinear form, a new superconvergence rate, which improves the existing result, is obtained.

Unfitted finite element for optimal control problem of the temperature in composite media with contact resistance

This paper presents a numerical method for the optimal control problem governed by the heat diffusion equation inside a composite medium. The contact resistance at the interface of constitute materials allows for jumps of the temperature field. The derivation process of the Karush-Kuhn-Tucher system is given by the formal Lagrange method. Due to the discontinuity of the temperature field, the standard linear finite element method cannot achieve optimal convergence when the uniform mesh is used. Therefore, the unfitted finite element method is applied to discrete the state equation required in the variational discretization approach. Optimal error estimates in the broken H1-norm and L2-norm for the control, state, and adjoint state are derived. Some numerical examples are provided to confirm the theoretical results.

Adaptive finite element methods for sparse PDE-constrained optimization

Abstract We propose and analyse reliable and efficient a posteriori error estimators for an optimal control problem that involves a nondifferentiable cost functional, the Poisson problem as state equation and control constraints. To approximate the solution to the state and adjoint equations we consider a piecewise linear finite element method, whereas three different strategies are used to approximate the control variable: piecewise constant discretization, piecewise linear discretization and the so-called variational discretization approach. For the first two aforementioned solution techniques we devise an error estimator that can be decomposed as the sum of four contributions: two contributions that account for the discretization of the control variable and the associated subgradient and two contributions related to the discretization of the state and adjoint equations. The error estimator for the variational discretization approach is decomposed only in two contributions that are related to the discretization of the state and adjoint equations. On the basis of the devised a posteriori error estimators, we design simple adaptive strategies that yield optimal rates of convergence for the numerical examples that we perform.

A finite element method of the self-consistent field theory on general curved surfaces

Block copolymers provide a wonderful platform in the study of soft condensed matter systems. Many fascinating ordered structures have been discovered in bulk and confined systems. Among various theories, the self-consistent field theory (SCFT) has been proven to be a powerful tool for studying the equilibrium ordered structures. Many numerical methods have been developed to solve the SCFT model. However, most of these focus on the bulk systems, and little work on the confined systems, especially on general curved surfaces. In this work, we developed a linear surface finite element method, which has a rigorous mathematical theory to guarantee numerical precision, to study the self-assembled phases of block copolymers on general curved surfaces based on the SCFT. Furthermore, to capture the consistent surface for a given self-assembled pattern, an adaptive approach to optimize the size of the general curved surface has been proposed. To demonstrate the power of this approach, we investigate the self-assembled patterns of diblock copolymers on several distinct curved surfaces, including five closed surfaces and an unclosed surface. Numerical results illustrate the efficiency of the proposed method. The obtained ordered structures are consistent with the previous results on standard surfaces, such as sphere and torus. More significantly, the proposed numerical framework can be applied to study the phase behaviors of block copolymers on general surfaces accurately.

Finite volume method for solving the stochastic Helmholtz equation

In this paper, we consider the linear finite volume method (FVM) for the stochastic Helmholtz equation, driven by white noise perturbed forcing terms in one-dimension. We first deduce the linear FVM for the deterministic Helmholtz problem. The dispersion equation is presented, and the error between the numerical wavenumber and the exact wavenumber is then analyzed. Comparisons between the linear FVM and the linear finite element method (FEM) are also made. The theoretical analysis and practical calculation indicate that the error of the linear FVM is half of that of the linear FEM. For the stochastic Helmholtz equation, convergence analysis and error estimates are given for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Numerical experiments are provided to examine our theoretical results.

Approximate solution of linear Volterra integro-differential equation by using cubic B-spline finite element method in the complex plane

So far, there are no any publications for solving and obtaining a numerical solution of Volterra integro-differential equations in the complex plane by using the finite element method. In this work, we use the linear B-spline finite element method (LBS-FEM) and cubic B-spline finite element method (CBS-FEM) for solving this equation in the complex plane. We also discuss the error and convergence of the method. Furthermore, we give some numerical examples to substantiate efficiency of the proposed method.

Anisotropic linear triangle finite element approximation for multi-term time-fractional mixed diffusion and diffusion-wave equations with variable coefficient on 2D bounded domain

A fully-discrete approximate scheme is established for the 2D multi-term time-fractional mixed diffusion and diffusion-wave equations with spatial variable coefficient by using linear triangle finite element method in space and classical L1 time-stepping method combined with Crank–Nicolson scheme in time. Then, the unconditionally stable analysis of the fully-discrete scheme is presented by employing some important lemmas. At the same time, both the spatial superclose property in H1-norm and convergence result in L2-norm are derived by skillfully dealing with numerical errors without any restrictions of time step τ and mesh size h. As a necessary way for obtaining the aimed numerical analysis, the relationship between the nonstandard projection operator Rh and the interpolation operator Ih of linear triangle finite element is introduced. Moreover, the global superconvergence is deduced by adopting interpolation postprocessing technique. Finally, numerical tests are provided to illustrate the efficiency and correctness of the theoretical results.

Galerkin FEM for a time-fractional Oldroyd-B fluid problem

We consider the numerical approximation of a generalized fractional Oldroyd-B fluid problem involving two Riemann-Liouville fractional derivatives in time. We establish regularity results for the exact solution which play an important role in the error analysis. A semidiscrete scheme based on the piecewise linear Galerkin finite element method in space is analyzed, and optimal with respect to the data regularity error estimates are established. Further, two fully discrete schemes based on convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are investigated and related error estimates for smooth and nonsmooth data are derived. Numerical experiments are performed with different values of the problem parameters to illustrate the efficiency of the method and confirm the theoretical results.

Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint

Abstract In this work we present numerical analysis for a distributed optimal control problem, with box constraint on the control, governed by a subdiffusion equation that involves a fractional derivative of order $\alpha \in (0,1)$ in time. The fully discrete scheme is obtained by applying the conforming linear Galerkin finite element method in space, L1 scheme/backward Euler convolution quadrature in time, and the control variable by a variational-type discretization. With a space mesh size $h$ and time stepsize $\tau $ we establish the following order of convergence for the numerical solutions of the optimal control problem: $O(\tau ^{\min ({1}/{2}+\alpha -\epsilon ,1)}+h^2)$ in the discrete $L^2(0,T;L^2(\varOmega ))$ norm and $O(\tau ^{\alpha -\epsilon }+\ell _h^2h^2)$ in the discrete $L^{\infty }(0,T;L^2(\varOmega ))$ norm, with any small $\epsilon>0$ and $\ell _h=\ln (2+1/h)$. The analysis relies essentially on the maximal $L^p$-regularity and its discrete analogue for the subdiffusion problem. Numerical experiments are provided to support the theoretical results.

Finite element evaluation of the effects of curvature in Lamb waves for composites structural health monitoring

Abstract This work presents the effect of the high curvature to thickness ratio on the characteristics of Lamb Waves propagating over the skins of composite structures. It is accessed how the curvature on composite skins affects the group velocity of the symmetrical (S0) and asymmetrical (A0) wave modes. It is also accessed the gradient of curvature effect, when the wave propagates from a flat to a curved part on the skin. The results are intended to be used for the improvement of structural health monitoring of wings and wind turbine blades. This is accomplished through dynamic explicit linear finite element method simulations of plates with flat and curved parts made of Eglass-epoxy bidirectional laminate. As the skin structures are often designed to withstand torsional loads, the fibers are aligned with 45 degrees from the leading edge (curved region) throughout the analysis. Several skins with different curvature to thickness ratios are generated and simulated. Results and trends are presented and can be used to improve the algorithms for damage detection on those structures. It is shown that the group velocity of the incoming waves change considerably with the presence of curvature, for both main modes of vibration.

A linear complete extended finite element method for dynamic fracture simulation with non-nodal enrichments

A linear complete extended finite element method for arbitrary dynamic crack is presented. In this method, strong and weak discontinuities are assigned to a set of non-nodal points on the interface, whereby the discontinuous functions across the interface are reproduced by extended interpolation. The enrichments are described to reproduce both the constants and linear functions on sides of the interface, which are critical for finite element convergence. A key feature of this method is that the enrichment descriptions and the finite element mesh are optimally uncoupled; the element nodes are not enriched facilitating the treatment of crack modeling in object-oriented programs. The enrichment variables are physically-based quantities which lead to a strong imposition of both the Dirichlet boundary conditions and the interface conditions. The convergence of the method is validated through static simulations from linear elastic fracture mechanics. The efficacy of the method for modeling dynamic crack propagation is demonstrated through two benchmark problems.

Two-Level Transport Methods with Independent Discretization

In this paper, we present two-level nonlinear iteration methods for the transport equation in 1D slab geometry approximated by means of the linear discontinuous finite element method (LDFEM). We develop transport schemes based on the quasidiffusion (QD) method in which the low-order QD (LOQD) equations are discretized by the linear continuous finite element method (LCFEM). This requires a mapping of the LCFEM low-order solution to the LDFEM high-order solution to define the scattering term. Several mappings are proposed and analyzed. Another proposed transport discretization scheme is based on the step characteristics for the transport equation and LCFEM for the LOQD equations. We also develop new nonlinear synthetic acceleration (NSA) methods based on the LCFEM discretization of the QD equation. To gain iterative stability, the NSA algorithms are combined with the nonlinear Krylov acceleration method. We present numerical results that demonstrate performance and basic properties of the proposed discretization schemes and iterative solution methods.

Numerical analysis of history-dependent variational–hemivariational inequalities with applications in contact mechanics

Abstract This paper is devoted to numerical analysis of history-dependent variational– hemivariational inequalities arising in contact problems for viscoelastic material. We introduce both temporally semi-discrete approximation and fully discrete approximation for the problem, where the temporal integration is approximated by a trapezoidal rule and the spatial variable is approximated by the finite element method. We analyze the discrete schemes and derive error bounds. The results are applied for the numerical solution of a quasistatic contact problem. For the linear finite element method, we prove that the error estimation for the numerical solution is of optimal order under appropriate solution regularity assumptions.

High-Precision Chebyshev Spectral to Approximate Laplace Eigenvalue Problem

In this paper, We focused on the Lapalce eigenvalue problem which solved by Chebyshev spectral based on Lagrange polynomial. The application background of this problem is mainly light film vibration, thermal magnetic radiation, and lattice vibration, etc. This paper used spectral theory to solve the Laplace eigenvalues with high-precision. Through compared we found that our method is better than the other linear finite element method, because Chebyshev spectral method is a global approximation, it has achieved superconvergence, spectral accuracy, and easier programming, we also analysis the error of this method and projecting the eigenvectors in this paper, we found that the eigenvalue vector was approximated by this method which has more stability and smoothly. At the same time, this method also provides a new and effective reference for solving Laplace eigenvalue problem.

Influence of electrode width of interdigital transducer on third-order nonlinearity of surface acoustic wave devices on 42°YX-LiTaO3 substrate

In this paper, we discuss the influence of the electrode width of an interdigital transducer on the third-order nonlinearity of surface acoustic wave (SAW) devices. First, an estimation technique of third-order nonlinear signals based on the linear finite element method is proposed, and the variation of nonlinear signal level with electrode width is estimated. Then, several one-port SAW resonators with different electrode widths are fabricated, and measured nonlinear signal levels are compared with simulation. As predicted by the numerical simulation, nonlinear signal levels became large with electrode width. However, harmonics takes a minimum at a certain electrode width. This tendency disagrees with the simulation. The variation of nonlinear coefficients is evaluated by numerical fitting for the measured data using the nonlinear signal simulator proposed by the authors. As the result, it is concluded that the generation mechanism is not limited to the acoustic strain in electrodes.

Discontinuous Finite Element Quasi-Diffusion Methods

In this paper, two-level methods for solving transport problems in one-dimensional slab geometry based on the quasi-diffusion (QD) method are developed. A linear discontinuous finite element method (LDFEM) is derived for the spatial discretization of the low-order QD (LOQD) equations. It involves special interface conditions at the cell edges based on the idea of QD boundary conditions (BCs). We consider different kinds of QD BCs to formulate the necessary cell-interface conditions. We develop two-level methods with independent discretization of the high-order transport equation and LOQD equations, where the transport equation is discretized using the method of characteristics and the LDFEM is applied to the LOQD equations. We also formulate closures that lead to the discretization consistent with a LDFEM discretization of the transport equation. The proposed methods are studied by means of test problems formulated with the method of manufactured solutions. Numerical experiments are presented demonstrating the performance of the proposed methods. We also show that the method with independent discretization has the asymptotic diffusion limit.

Approximating coupled hyperbolic–parabolic systems arising in enhanced drug delivery

In this paper we study a system of partial differential equations defined by a hyperbolic equation and a parabolic equation. The convective term of the parabolic equation depends on the solution and eventually on the gradient of the solution of the hyperbolic equation. This system arises in the mathematical modeling of several physical processes as for instance ultrasound enhanced drug delivery. In this case the propagation of the acoustic wave, which is described by a hyperbolic equation, induces an active drug transport that depends on the acoustic pressure. Consequently the drug diffusion process is governed by a hyperbolic and a convection–diffusion equation.Here, we propose a numerical method that allows us to compute second-order accurate approximations to the solution of the hyperbolic and the parabolic equation. The method can be seen as a fully discrete piecewise linear finite element method or as a finite difference method. The convergence rates for both approximations are unexpected. In fact we prove that the error for the approximation of the pressure and concentration is of second-order with respect to discrete versions of the H1-norm and L2-norm, respectively. Numerical experiments illustrating the theoretical estimates are also presented. A simplified model for ultrasound enhanced drug transport is also discussed.