On the Numerical Solutions of Helmholtz’s Equation by the Finite Element Method

1980 ◽  
Vol 17 (5) ◽  
pp. 681-686 ◽  
Author(s):  
A. K. Aziz ◽  
A. Werschulz
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Solutions ◽  
The Finite Element Method ◽  
Element Method

Linearization of the finite element method for gradient flows by Newton’s method

2020 ◽  
Author(s):  
Georgios Akrivis ◽  
Buyang Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Newton’S Method ◽  
Newton's Method ◽  
Numerical Solutions ◽  
Resolvent Estimate ◽  
Optimal Order ◽  
Gradient Flows ◽  
The Finite Element Method ◽  
Element Method

Abstract The implicit Euler scheme for nonlinear partial differential equations of gradient flows is linearized by Newton’s method, discretized in space by the finite element method. With two Newton iterations at each time level, almost optimal order convergence of the numerical solutions is established in both the $L^q(\varOmega )$ and $W^{1,q}(\varOmega )$ norms. The proof is based on techniques utilizing the resolvent estimate of elliptic operators on $L^q(\varOmega )$ and the maximal $L^p$-regularity of fully discrete finite element solutions on $W^{-1,q}(\varOmega )$.

Increasing the Solution Accuracy in the Numerical Modeling of Boundary Value Problems Using Finite Element Method Based on Hankel Shape Functions

2019 ◽  
Vol 11 (07) ◽  
pp. 1950062
Author(s):  
S. Farmani ◽  
M. Ghaeini-Hessaroeyeh ◽  
S. Hamzehei-Javaran
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Value Problems ◽  
Numerical Solutions ◽  
Boundary Value ◽  
The Finite Element Method ◽  
Shape Functions ◽  
Element Approach ◽  
Solution Accuracy ◽  
Element Method

A new finite element approach is developed here for the modeling of boundary value problems. In the present model, the finite element method (FEM) is reformulated by new shape functions called spherical Hankel shape functions. The mentioned functions are derived from the first and second kind of Bessel functions that have the properties of both of them. These features provide an improvement in the solution accuracy with number of elements which are equal or lower than the ones used by the classic FEM. The efficiency and accuracy of the suggested model in the potential problems are examined by several numerical examples. Then, the obtained results are compared with the analytical and numerical solutions. The comparisons indicate the high accuracy of the present method.

scholarly journals Finite element modeling of fluid flow in fractured porous media using unified approach

2020 ◽  
Vol 43 (1) ◽  
pp. 13-22
Author(s):  
Hai-Bang Ly ◽  
Hoang-Long Nguyen ◽  
Minh-Ngoc Do
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Porous Media ◽  
Fluid Flow ◽  
Numerical Solutions ◽  
Fractured Porous Media ◽  
Flow In Porous Media ◽  
Soil Science ◽  
The Finite Element Method ◽  
Element Method

Understanding fluid flow in fractured porous media is of great importance in the fields of civil engineering in general or in soil science particular. This study is devoted to the development and validation of a numerical tool based on the use of the finite element method. To this aim, the problem of fluid flow in fractured porous media is considered as a problem of coupling free fluid and fluid flow in porous media or coupling of the Stokes and Darcy equations. The strong formulation of the problem is constructed, highlighting the condition at the free surface between the Stokes and Darcy regions, following by the variational formulation and numerical integration using the finite element method. Besides, the analytical solutions of the problem are constructed and compared with the numerical solutions given by the finite element approach. Both local properties and macroscopic responses of the two solutions are in excellent agreement, on condition that the porous media are sufficiently discretized by a certain level of finesse. The developed finite element tool of this study could pave the way to investigate many interesting flow problems in the field of soil science.

Use of the Finite-Element Method in the Solution of Diffusion-Convection Equations

1971 ◽  
Vol 11 (02) ◽  
pp. 139-144 ◽  
Author(s):  
Y.M. Shum
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heat Conduction ◽  
Variational Principle ◽  
Numerical Solutions ◽  
Transient Heat Conduction ◽  
The Finite Element Method ◽  
One Dimensional ◽  
Diffusion Convection ◽  
Element Method

Abstract A variational principle can be applied to the transient heat conduction equation with heat-flux boundary conditions. The finite-element method is employed to reduce the continuous spatial solution into a finite number of time-dependent unknowns. From previous work, it was demonstrated that the method can readily be applied to solve problems involving either linear or nonlinear boundary conditions, or both. In this paper, with a slight modification of the solution technique, the finite-element method is shown to be applicable to diffusion-convection equations. Consideration is given to a one-dimensional transport problem with dispersion in porous media. Results using the finite-element method are compared with several standard finite-difference numerical solutions. The finite-element method is shown to yield satisfactory solutions. Introduction The problem of finding suitable numerical approximations for equations describing the transport of heat (or mass) by conduction (or diffusion) and convection simultaneously has been of interest for some time. Equations of this type, which will be called diffusion-convection equations, arise in describing many diverse physical processes. Of particular interest to petroleum engineers is the classical equation describing the process by which one miscible fluid displaces another in a one-dimensional porous medium. Many authors have presented numerical solutions to this rather simple presented numerical solutions to this rather simple diffusion-convection problem using standard finite-difference methods, method of characteristics, and variational methods. In this paper another numerical method is employed. A finite-element method in conjunction with a variational principle for transient heat conduction analysis is briefly reviewed. It is appropriate here to mention the recent successful application of the finite-element method to solve transient heat conduction problems involving either linear, nonlinear, or both boundary conditions. The finite-element method was also applied to transient flow in porous media in a recent paper by Javandel and Witherspoon. Prime references for the method are the papers by Gurtin and Wilson and Nickell. With a slight modification of the solution procedure for treating the convective term as a source term in the transient heat conduction equation, the method can readily be used to obtain numerical solutions of the diffusion-convection equation. Consideration is given to a one-dimensional mass transport problem with dispersion in a porous medium. Results using the finite-element method yield satisfactory solutions comparable with those reported in the literature. A VARIATIONAL PRINCIPLE FOR TRANSIENT HEAT CONDUCTION AND THE FINITE-ELEMENT METHOD A variational principle can be generated for the transient conduction or diffusion equation. Wilson and Nickell, following Gurtin's discussion of variational principles for linear initial value problems, confirmed that the function of T(x, t) that problems, confirmed that the function of T(x, t) that leads to an extremum of the functional...........(1) is, at the same time, the solution to the transient heat conduction equation SPEJ P. 139

Comparison of Analytical Solutions with Finite Element Solutions for Ultimate Bearing Capacity of Strip Footings

2013 ◽  
Vol 353-356 ◽  
pp. 3294-3303
Author(s):  
Zi Hang Dai ◽  
Xiang Xu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Bearing Capacity ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Slip Surface ◽  
Ultimate Bearing Capacity ◽  
The Finite Element Method ◽  
Strip Footings ◽  
Element Method

The finite element method is used to compute the ultimate bearing capacity of a fictitious strip footing resting on the surface of c-φ weightless soils and a real strip footing buried in the c-φ soils with weight. In order to compare the numerical solutions with analytical solutions, the mainly existing analytical methods are briefly introduced and analyzed. To ensure the precision, most of analytical solutions are obtained by the corresponding formulas rather than table look-up. The first example shows that for c-φ weightless soil, the ABAQUS finite element solution is almost identical to the Prandtls closed solutions. Up to date, though no closed analytical solution is obtained for strip footings buried in c-φ soils with weight, the numerical approximate solutions obtained by the finite element method should be the closest to the real solutions. Apparently, the slip surface disclosed by the finite element method looks like Meyerhofs slip surface, but there are still some differences between the two. For example, the former having an upwarping curve may be another log spiral line, which begins from the water level of footing base to ground surface rather than a straight line like the latter. And the latter is more contractive than the former. Just because these reasons, Meyerhofs ultimate bearing capacity is lower than that of the numerical solution. Comparison between analytical and numerical solutions indicates that they have relatively large gaps. Therefore, finite element method can be a feasible and reliable method for computations of ultimate bearing capacity of practical strip footings.

Rock Bit Cutter Tooth Design and Analysis

1983 ◽  
Vol 105 (2) ◽  
pp. 138-144
Author(s):  
G. C. Feng ◽  
P. W. Schumacher
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Solutions ◽  
Plastic Region ◽  
Finite Element Models ◽  
The Finite Element Method ◽  
Stress Concentrations ◽  
Property Changes ◽  
Rock Bit ◽  
Element Method

The finite element method was used to analyze the stresses in a rock bit cutter tooth under applied loads. Detailed finite element models were constructed to allow accurate descriptions of the structural material properties of a tooth design which includes carburization and hardmetal application. Numerical solutions were obtained for investigating stress concentrations and plastic region growths in a cutter tooth when loaded beyond its yield point. It was found that the finite element method can be a powerful tool for improving the performance of a cutter tooth. Design objectives can be achieved through parameteric studies of stress level variations against geometry and material property changes on a computer system.

Numerical Verification of Model Multibody Systems

2013 ◽  
Vol 486 ◽  
pp. 239-244 ◽  
Author(s):  
Martin Svoboda
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Model ◽  
Multibody Systems ◽  
Numerical Solutions ◽  
Rigid Bodies ◽  
Laboratory Model ◽  
Numerical Verification ◽  
The Finite Element Method ◽  
Element Method

The article deals with the influence of manufacturing and geometric asymmetry on the vertical oscillation of symmetric and asymmetric systems consisting of rigid bodies linked flexibly with different kinematic excitation. The solution was performed using experimental and numerical analysis (with application of oscillation of vehicles and flexibly coupled machines). Numerical solutions were carried out by finite element method (FEM) on a simple model and experimental solutions on laboratory model of a mechanical system. The aim of the work was to create a numerical model and its solution using the finite element method. The experimental solution was used to verify the numerical model.

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