Finite Element Method Of Lines Research Articles

Neural networks-based spring element for second-order analysis of pile-supported structures with nonlinear soil- structure interaction

This paper proposes a novel structural analysis approach, the neural networks-based spring element (NNSE) method, to synergize machine learning (ML) techniques with the line finite element method (LFEM) for the second-order analysis method of pile-supported structures. Traditional LFEM, widely used in upper structure design, showcases limitations in efficiently modeling complex Soil-Structure Interaction (SSI) along piles since it requires dense element mesh for accuracy. Conversely, ML offers a mesh-free alternative for analyzing single piles but struggles in simulating pile-supported structures as the training sample collection for large-scale problems might be unbearable. This paper addresses these challenges by proposing a new analysis framework to utilize the neural network (NN) model, which only describes structural responses of single piles, for the simulation of entire pile-supported structures. In the proposed method, the NN model is not directly used for structural analysis but employed to formulate a new spring element named the NNSE to model single piles in pile-supported structures. This NNSE can be seamlessly implemented within the existing LFEM framework to analyze pile-supported structures, eliminating the dense mesh requirement for single piles and thereby significantly improving the computational efficiency. Extensive examples are provided to verify the effectiveness of the proposed method, indicating its potential in promoting the second-order analysis method to the design of pile-supported structures.

Efficient numerical implementation for nonlinear analysis of pile-supported structures during soil liquefaction through adaptive pile element formulations

The structural behavior of pile-supported structures during soil liquefaction is sometimes insufficiently investigated properly due to the inefficiency in the existing line finite element method (LFEM), which is widely used in practice. The traditional LFEM to simulate pile response requires dense element mesh for accurate results, as it simplifies the continuous surrounding soil as discrete spring elements, resulting in high computational costs, especially for pile-supported structures with a significant number of piles. Additionally, analyzing pile-supported structures in different stages of soil liquefaction requires repetitive updating of p-y curves and tedious starting-over iteration processes. To address these issues, an innovative line element formulation named the pile element formulation is employed and improved to adaptively simulate the surrounding soil conditions at different stages during liquefaction within the element formulation, eliminating the need to model discrete soil springs and significantly improving computational efficiency. The adaptive pile element formulation is then implemented within a refined Newton-Raphson iteration procedure that is proposed for a more effective nonlinear analysis process to consider soil liquefaction without repetitive iteration processes from starting over. Through several examples, the computational efficiency of the proposed method is validated, which requires only 20% of the computational cost compared to traditional LFEM while still providing accurate results, highlighting the application potential of the present study.

Adaptive Finite Element Method of Lines with Local Mesh Refinement in Maximum Norm Based on Element Energy Projection Method

The reliable and efficient self-adaptive analysis is a modern goal of various numerical computations. Most adaptivity methods, however, adopt energy norm to measure errors, which may not be the most natural and convenient means, e.g., for problems with locally singular gradient of displacement. Based on the Element Energy Projection (EEP) super-convergent technique in the Finite Element Method of Lines (FEMOL) which is a general and powerful semi-discrete method, reliable error estimates of displacements in maximum norm can be obtained anywhere on the FEMOL mesh and hence adaptive FEMOL by maximum norm becomes feasible. However, to tackle singularity problems effectively and efficiently, an automatic and flexible local mesh refinement strategy is required to generate meshes of high quality for more efficient adaptive FEMOL analysis. Taking the two-dimensional Poisson equation as the model problem, the paper firstly introduces the FEMOL and EEP methods with interface sides resulting from local mesh refinement. Then a local mesh refinement strategy and corresponding adaptive algorithm are presented. The numerical results given show that the proposed adaptive FEMOL with local mesh refinement can produce displacement solutions satisfying the specified tolerances in maximum norm and the adaptively-generated meshes reasonably reflect the local difficulties inherent in the physical problems without much redundant accuracy.

A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems

AbstractThis article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time. The method presented here uses immersed finite element (IFE) functions for the discretization in spatial variables that can be carried out over a fixed mesh (such as a Cartesian mesh if desired), and this feature makes it possible to reduce the parabolic equation to a system of ordinary differential equations (ODE) through the usual semi-discretization procedure. Therefore, with a suitable choice of the ODE solver, this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured (Cartesian) mesh. Numerical examples are presented to demonstrate features of this new method.

Computation of stress intensity factors by the sub-region mixed finite element method of lines

Based on the sub-region generalized variational principle, a sub-region mixed version of the newly-developed semi-analytical ‘finite element method of lines’ (FEMOL) is proposed in this paper for accurate and efficient computation of stress intensity factors (SIFs) of two-dimensional notches/cracks. The circular regions surrounding notch/crack tips are taken as the complementary energy region in which a number of leading terms of singular solutions for stresses are used, with the sought SIFs being among the unknown coefficients. The rest of the arbitrary domain is taken as the potential energy region in which FEMOL is applied to obtain approximate displacements. A mixed system of ordinary differential equations (ODEs) and algebraic equations is derived via the sub-region generalized variational principle. A singularity removal technique that eliminates the stress parameters from the mixed equation system eventually yields a standard FEMOL ODE system, the solution of which is no longer singular and is simply and efficiently obtained using a standard general-purpose ODE solver. A number of numerical examples, including bi-material notches/cracks in anti-plane and plane elasticity, are given to show the generally excellent performance of the proposed method.

Stress intensity factors calculation in anti-plane fracture problem by orthogonal integral extraction method based on FEMOL

For an anti-plane problem, the differential operator is self-adjoint and the corresponding eigenfunctions belong to the Hilbert space. The orthogonal property between eigenfunctions (or between the derivatives of eigenfunctions) of anti-plane problem is exploited. We developed for the first time two sets of radius-independent orthogonal integrals for extraction of stress intensity factors (SIFs), so any order SIF can be extracted based on a certain known solution of displacement (an analytic result or a numerical result). Many numerical examples based on the finite element method of lines (FEMOL) show that the present method is very powerful and efficient.

Cracking analysis of fracture mechanics by the finite element method of lines (FEMOL)

The Finite Element Method of Lines (FEMOL) is a semi-analytic approach and takes a position between FEM and analytic methods. First, FEMOL in Fracture Mechanics is presented in detail. Then, the method is applied to a set of examples such as edge-crack plate, the central-crack plate, the plate with cracks emanating from a hole under tensile or under combination loads of tensile and bending. Their dimensionless stress distribution, the stress intensify factor (SIF) and crack opening displacement (COD) are obtained, and comparison with known solutions by other methods are reported. It is found that a good accuracy is achieved by FEMOL. The method is successfully modified to remarkably increase the accuracy and reduce convergence difficulties. So it is a very useful and new tool in studying fracture mechanics problems.

Efficient hybrid scheme of finite element method of lines for modelling computational electromagnetic problems

AbstractA hybrid scheme called finite element method of lines is proposed and described for modelling and analysis of generalized computational electromagnetic problems with emphasis on a number of irregular waveguide examples. This new technique is developed by combining a finite element method with a method of lines so that it can handle not only irregular composite geometry but also maintain high accuracy enjoyed by semi‐analytical procedures. Analytical and numerical algorithmic building blocks of this new scheme are discussed in detail such as geometry discretization, element mapping, element trial functions, reformulation and computational issues of non‐linear ordinary differential equations. Our results show that this new technique is able to efficiently solve complex problems as compared with the conventional method of lines. Copyright © 2004 John Wiley & Sons, Ltd.

A posteriori error estimates with the finite element method of lines for a Sobolev equation

AbstractA posteriori error estimates for semidiscrete finite element methods for a nonlinear Sobolev equation are considered. The error estimates are obtained by solving local nonlinear or linear pseudo‐parabolic equations for corrections to the solution on each element. The ratios of these estimates and the true errors are proved to converge to 1, implying that the estimates can be used as indicators in adaptive schemes for the problem. Numerical results underline our theoretical results. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems

The subject-matter is the analysis of the discontinuous Galerkin finite element method of lines applied to a linear nonstationary convection–diffusion–reaction problem. In the contrary to the standard FEM the requirement of the conforming properties is omitted. The discretization is carried out with respect to space variables, whereas time remains continuous. In the discontinuous Galerkin discretization, the nonsymmetric stabilization of diffusion terms combined with interior and boundary penalty is applied. In the evaluation of fluxes the idea of upwinding is used. This allows to obtain an optimal error estimate, also verified by numerical experiments.

Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems

The subject-matter is the analysis of the discontinuous Galerkin finite element method of lines applied to a linear nonstationary convection–diffusion–reaction problem. In the contrary to the standard FEM the requirement of the conforming properties is omitted. The discretization is carried out with respect to space variables, whereas time remains continuous. In the discontinuous Galerkin discretization, the nonsymmetric stabilization of diffusion terms combined with interior and boundary penalty is applied. In the evaluation of fluxes the idea of upwinding is used. This allows to obtain an optimal error estimate, also verified by numerical experiments.

Singularity Intensity Factor Calculation in Plates and Shells Based on the Singular Line Mapping Technique in FEMOL

In this paper, a brief review of the singular line mapping technique in finite element method of lines (FEMOL) for plate and shell crack/notch problems is given. By using the singular line mapping technique, the solution property is changed, the computation is speeded up and the accuracy is improved. The displacement and its derivative can be obtained accurately after an accurate singular line mapping, and the lowest stress intensity factor can be solved from the derivative. Some numerical examples are given to show the general excellent performance.

A posteriori error estimates and grid adjustment for a nonlinear parabolic equation

The paper is concerned with a posteriori error estimates needed for the adaptive construction of a space grid in solving an initial-boundary value one-dimensional problem for a nonlinear parabolic partial differential equation by the method of lines. A posteriori error indicators are introduced and studied for the semidiscrete case, and the statement on their convergence rate is presented. Under certain conditions, it adds some more results to those of Moore [SIAM J. Numer. Anal. 31 (1994) 149–164] who treats only a linear equation in the semidiscrete case. Complete statements including proofs will appear as a paper [K. Segeth, A posteriori error estimation with the finite element method of lines for a nonlinear parabolic equation in one space dimension, to appear]. A simple numerical example illustrates the capabilities of a posteriori error estimates when used for grid adjustments in the finite element method of lines. The example fully confirms the theoretical statement.

A novel finite-element method of lines to analyze electromagnetic problems

A novel finite-element method of lines to analyze electromagnetic problems

Grid adjustment for parabolic systems based on a posteriori error estimates

The construction of a one-dimensional space grid for solving a system of parabolic partial differential equations by the finite element method of lines is considered in the paper. In particular, the approach based on a posteriori error indicators and error estimators is studied. Two a posteriori error estimators for the approximate solution are presented together with statements on their rate of convergence. Results obtained for a numerical example are shown in conclusion.

A posteriori error estimates for parabolic differential systems solved by the finite element method of lines

Systems of parabolic differential equations are studied in the paper. Two a posteriori error estimates for the approximate solution obtained by the finite element method of lines are presented. A statement on the rate of convergence of the approximation of error by estimator to the error is proved.

A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems

Consider the solution of one-dimensional linear initial-boundary value problems by a finite element method of lines using a piecewiseP th -degree polynomial basis. A posteriori estimates of the discretization error are obtained as the solutions of either local parabolic or local elliptic finite element problems using piecewise polynomial corrections of degreep+1 that vanish at element ends. Error estimates computed in this manner are shown to converge in energy under mesh refinement to the exact finite element discretization error. Computational results indicate that the error estimates are robust over a wide range of mesh spacings and polynomial degrees and are, furthermore, applicable in situations that are not supported by the analysis.

Extension of the E-plane scanning range in large microstrip arrays by substrate modification

The effect of the substrate structure modification on the E-plane scanning performance in large microstrip arrays is studied with the aid of a two-dimensional model. The goal is to alleviate scan blindness effects. The finite-element method of lines is used to solve the electromagnetic equations, thus demonstrating its applicability to open, periodic structures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Finite element method of lines analysis of open periodic structures

The applicability of the finite element method of lines to the analysis of open, periodic structures in demonsrated. To validate the method and study its characteristics, the problem of transverse electric (TE) plane wave scattering from a metallic waveguide array is considered. Convergence of the solution is ivestigated both as a function of the mesh density and the order of the polynomial approximation used in the construction of basis functions.

Second-order finite element approximations and a posteriori error estimation for two-dimensional parabolic systems

We discuss an adaptive local refinement finite element method of lines for solving vector systems of parabolic partial differential equations on two-dimensional rectangular regions. The partial dif...