scholarly journals Description and implementation of an algebraic multigrid preconditioner for H1-conforming finite element schemes

Uniciencia ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 55-81
Author(s):  
Helen Guillén-Oviedo ◽  
Jeremías Ramírez-Jiménez ◽  
Esteban Segura-Ugalde ◽  
Filánder Sequeira-Chavarría
Keyword(s):  
Finite Element ◽  
Numerical Experiments ◽  
Mesh Size ◽  
Algebraic Multigrid ◽  
The Finite Element Method ◽  
Conforming Finite Element ◽  
Finite Element Schemes ◽  
Pcg Method ◽  
Continuous Problem ◽  
Number Of Iterations

This paper presents detailed aspects regarding the implementation of the Finite Element Method (FEM) to solve a Poisson’s equation with homogeneous boundary conditions. The aim of this paper is to clarify details of this implementation, such as the construction of algorithms, implementation of numerical experiments, and their results. For such purpose, the continuous problem is described, and a classical FEM approach is used to solve it. In addition, a multilevel technique is implemented for an efficient resolution of the corresponding linear system, describing and including some diagrams to explain the process and presenting the implementation codes in MATLAB®. Finally, codes are validated using several numerical experiments. Results show an adequate behavior of the preconditioner since the number of iterations of the PCG method does not increase, even when the mesh size is reduced.

Stratification and Buoyancy Effect of Heat Transportation in Magnetohydrodynamics Micropolar Fluid Flow Passing Over a Porous Shrinking Sheet Using the Finite Element Method

2019 ◽  
Vol 8 (8) ◽  
pp. 1640-1647 ◽  
Author(s):  
Shahid Ali Khan ◽  
Yufeng Nie ◽  
Bagh Ali
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Solution ◽  
Mesh Size ◽  
Stratified Medium ◽  
Shrinking Sheet ◽  
The Finite Element Method ◽  
Nonlinear Odes ◽  
Micropolar Fluid Flow ◽  
Element Method

The current study investigates the numerical solution of steady heat transportation in magnetohydrodynamics flow of micropolar fluids over a porous shrinking/stretching sheet with stratified medium and buoyancy force. Based on similarity transformation, the partial differential governing equations are assimilated into a set of nonlinear ODEs, which are numerically solved by the finite element method. All obtained unknown functions are discussed in detail after plotting the numerical results against different arising thermophysical parameters namely, suction, magnetic, stratification, heat source, and buoyancy parameter. Under the limiting case, the numerical solution of the velocity and temperature is compared with present work. Better consistency between the two sets of solutions was determined. To verify the convergence of the numerical solution, the calculation is made by reducing the mesh size. The present study finds applications in materials processing and demonstrates convergence characteristics for the finite element method code.

GENERAL BUCKLING ANALYSIS OF STEEL BUILT-UP COLUMNS USING FINITE ELEMENT MODELLING / PLIENINIŲ SPRAGOTŲJŲ KOLONŲ BENDROJO PASTOVUMO SKAIČIAVIMAS, MODELIUOJANT UŽDAVINĮ BAIGTINIAIS ELEMENTAIS

2012 ◽  
Vol 4 (1) ◽  
pp. 1-6
Author(s):  
Vaidotas Šapalas ◽  
Gintas Šaučiuvėnas ◽  
Arūnas Komka
Keyword(s):  
Finite Element ◽  
Finite Element Modelling ◽  
Numerical Experiments ◽  
Fem Analysis ◽  
Correction Factors ◽  
The Finite Element Method ◽  
Buckling Modes ◽  
Element Modelling ◽  
Eurocode 3 ◽  
Loaded Column

The paper investigates the general buckling of an axially loaded column using the finite element method with different slenderness ratios of axes x-x and z-z. The paper deals with three different modes of buckling. The conducted numerical experiments have suggested correction factors and appropriate buckling modes of the built-up columns. The obtained modelling results were compared with data on analytical calculations made according to Lithuanian national codes STR and Eurocode 3. The FEM analysis of the builtup column has showed that both codes (STR and EC3) are giving safe enough results for a considered type of conditions for column support. Santrauka Straipsnyje nagrinėjami plieninių spragotųjų kolonų elgsenos ypatumai, atsižvelgiant į skirtingas STR 2.05.08:2005 ir EC3-1-1 metodikas. Didžiausią susidomėjimą kelia faktas, kad, taikant EC3 metodiką, nėra nagrinėjama spragotosios kolonos kluptis apie didesnio standumo x-x ašį (1 pav.). Naudojantis turima ir kitų autorių patirtimi apžvelgta spragotosios centriškai gniuždomos kolonos elgsena, siekiant nustatyti jos klumpamąją galią, kai kolonos liauniai yra didesnio standumo, o mažesnio standumo plokštumoje yra skirtingi. Nagrinėtos trys skaičiuotinės situacijos: 1) spragotoji kolona idealiai tiesi, o abi kolonos juostos perima vienodas ašines jėgas (STR2.05.08:2005 prielaida); 2) spragotoji kolona tiesi, bet kolonos juostos ašinės jėgos padidintos dėl pradinio kolonos nuokrypio nuo tiesiosios ašies ir papildomo lenkiamojo momento (EC 3-1-1 prielaida); 3) kolona su pradiniu nuokrypiu, o ašinės juostų apkrovos yra vienodos (EC3-1-1 prielaida). Pradiniai modeliavimo duomenys pateikti 1 lentelėje. Kolonos juostos modeliuotos plokštelės tipo baigtinais elementais, o tinklelio strypai – strypiniais. Kolonos įtvirtinimo sąlygos ir skaičiuojamieji ilgiai pateikti 2 pav. Atlikus skaitinį modeliavimą gauti kolonos bendrojo klupumo pataisos koeficientai (2 lentelė) ir kolonos klupumo pavidalai (5 ir 6 pav.). Kaip matyti iš 7 pav., skaitinio modeliavimo rezultatai 2-uoju ir 3-iuoju atvejais yra skirtingi, nors turėtų būti vienodi modeliuojant pagal EC3 prielaidas. Galima teigti, kad antruoju atveju (spragotoji kolona tiesi, bet kolonos juostų ašinės jėgos padidintos dėl pradinio kolonos nuokrypio ir papildomo lenkiamojo momento) gaunami tikslesni rezultatai. Skirtumas tarp 1-ojo (STR) ir 2-ojo atvejo (EC3) nėra didelis; nuo 22% iki 30% (9 pav.). Kai kolonos liaunis λ x=35, pataisos koeficientas yra 2,11, naudojant EC3, ir 1,78, taikant STR metodą. Nedidelio liaunio kolonų bendrojo klupumo atsarga, naudojant EC3 prielaidas, yra 30% didesnė nei taikant STR metodą. Kai liaunis λ x=200, pataisos koeficientas yra 1,16 pagal EC3 ir 1,38 pagal STR metodiką. Liaunų kolonų klupumo atsarga pagal EC3 yra 23% mažesnė nei pagal STR metodą. 1-uoju ir 2-uoju atveju (1 lentelė) kolonų kluptis įvyko iš plokstumos apie z-z ašį (6 pav.), nes šioje plokštumoje kolonų liaunis didesnis. Kolonų klupumo pavidalas atitinka analitinius skaičiavimus tiek STR, tiek EC3 metodu. 3-uoju ir 4-uoju atveju (1 lentelė) kolonos klupo apie x-x ašį (5 pav.), nes šioje plokštumoje kolonos liaunis gerokai didesnis (2 pav.). Klupumo pavidalas atitinka analitinius skaičiavimus pagal STR metodiką. Taikant EC3 metodiką kolona turėjo klupti iš plokštumos, t. y. apie z-z ašį. Šis klupumo pavidalas taip pat buvo pasiektas, tačiau vėliau (žr. pataisos koeficientus 2 lentelėje.). Taip yra todėl, kad pagal EC3 metodiką tiesiog nereikalauja visos kolonos pastovumo tikrinti apie x-x ašį. Atlikus skaitinius modeliavimus galima teigti, kad abu metodai STR ir EC3 yra saugūs (pataisos koeficientai visada didesni už vienetą) duotomis kolonos galų įtvirtinimo sąlygomis. Tik mažo liaunio kolonų λx≤80 didesnė atsarga gauta STR metodu, o liaunų kolonų, kai λx>80 didesnė atsarga gauta taikant EC3 metodą.

Finite Element Solution of Inertia Influenced Flow in Thin Fluid Films

2007 ◽  
Vol 129 (4) ◽  
pp. 876-886 ◽  
Author(s):  
Noël Brunetière ◽  
Bernard Tournerie
Keyword(s):  
Finite Element ◽  
Numerical Model ◽  
Fluid Flows ◽  
Mesh Size ◽  
The Finite Element Method ◽  
Fluid Films ◽  
Inertia Effects ◽  
Continuity Equations ◽  
Numerical Precision ◽  
Incompressible Fluid Flows

The aim of this paper is to present a numerical model to compute laminar, turbulent, and transitional incompressible fluid flows in thin lubricant films where inertia effects cannot be neglected. For this purpose, an averaged inertia method is used. A numerical scheme based on the finite element method is presented to solve simultaneously the momentum and continuity equations. The numerical model is then validated by confronting it with previously published analytical, experimental, and numerical results. Particular attention is devoted to analyzing the numerical conservation of mass and momentum. The influence of mesh size on numerical precision is also analyzed. Finally, the model is applied to a misaligned hydrostatic seal. These seals operate with a substantial leakage flow, where nonlaminar phenomena occur. The influence of inertia and misalignment of the faces on the seal behavior is analyzed through a comparison with an inertialess solution. Significant differences are observed for high values of the tilt angle when the flow is nonlaminar. Inertia effects increase when the flow is laminar.

ELECTROMAGNETIC AND ELECTRICAL MODELING BY THE FINITE ELEMENT METHOD

Geophysics ◽  
1971 ◽  
Vol 36 (1) ◽  
pp. 132-155 ◽  
Author(s):  
J. H. Coggon
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Variational Equation ◽  
Mesh Size ◽  
Conducting Layer ◽  
Texture Element ◽  
Two Dimensional ◽  
The Finite Element Method ◽  
Source Excitation ◽  
Element Method

Application of the finite element method to the solution of physical problems is based on minimization of energy; in the present case electromagnetic energy is minimized. Representation of a volume of space by a number of finite elements and description of field or potential distribution by a finite set of unknown values make it possible to replace the energy variational equation by matrix equations. It is shown that a solution for secondary rather than total field quantities can be obtained directly. Such a procedure has several advantages. Approximations are involved in using non‐infinitesimal elements and finite meshes of elements. It is usually necessary to pay more attention to mesh size than texture (element size). Examples of induced polarization anomalies over two‐dimensional models illustrate effects of topography and of a highly conducting layer above bodies of polarizable material. Computed electromagnetic anomalies of two‐dimensional structures, with line source excitation, include the effects of adjacent conductors and magnetic conductors set in a less conductive half‐space.

scholarly journals Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

2021 ◽  
Vol 47 (5) ◽  
Author(s):  
I. G. Graham ◽  
O. R. Pembery ◽  
E. A. Spence
Keyword(s):  
Finite Element ◽  
Dirichlet Problem ◽  
Helmholtz Equation ◽  
Uncertainty Quantification ◽  
Numerical Experiments ◽  
Helmholtz Equations ◽  
Independent Number ◽  
Theoretical Evidence ◽  
Number Of Iterations ◽  
Quantities Of Interest

AbstractThis paper analyses the following question: let Aj, j = 1,2, be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations ∇⋅ (Aj∇uj) + k2njuj = −f. How small must $\|A_{1} -A_{2}\|_{L^{q}}$ ∥ A 1 − A 2 ∥ L q and $\|{n_{1}} - {n_{2}}\|_{L^{q}}$ ∥ n 1 − n 2 ∥ L q be (in terms of k-dependence) for GMRES applied to either $(\mathbf {A}_1)^{-1}\mathbf {A}_2$ ( A 1 ) − 1 A 2 or A2(A1)− 1 to converge in a k-independent number of iterations for arbitrarily large k? (In other words, for A1 to be a good left or right preconditioner for A2?) We prove results answering this question, give theoretical evidence for their sharpness, and give numerical experiments supporting the estimates. Our motivation for tackling this question comes from calculating quantities of interest for the Helmholtz equation with random coefficients A and n. Such a calculation may require the solution of many deterministic Helmholtz problems, each with different A and n, and the answer to the question above dictates to what extent a previously calculated inverse of one of the Galerkin matrices can be used as a preconditioner for other Galerkin matrices.

scholarly journals Numerical Analysis and Experimental Verification of Eigenfrequencies of Overhead ACSR Conductor

2020 ◽  
Vol 5 (4) ◽  
pp. 116-121
Author(s):  
Juraj Hrabovský ◽  
Roman Gogola ◽  
Vladimír Goga ◽  
František Janíček
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Modal Analysis ◽  
Cross Section ◽  
Material Properties ◽  
Numerical Experiments ◽  
Volume Element ◽  
Experimental Measurements ◽  
The Finite Element Method ◽  
Fgm Beam

<span lang="EN-GB">This contribution deals with the modal analysis of ACSR conductor using the finite element method (FEM) and experimental measurements of eigenfrequencies. In numerical experiments for the modelling of the conductor the material properties of the chosen conductor cross-section are homogenized by the </span><span lang="EN-US">Representative</span><span lang="EN-GB"> Volume Element (RVE) method. The spatial modal analysis of the power line is carried out by means of our new 3D FGM beam finite element and by standard beam finite element of the commercial software ANSYS. Experimental measurements are also carried out for verification of the numerical calculation accuracy.</span>

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