On convergence of the penalty method for a static unilateral contact problem with nonlocal friction in electro-elasticity

2015 ◽  
Vol 27 (1) ◽  
pp. 1-22 ◽  
Author(s):  
El-H. BENKHIRA ◽  
El-H. ESSOUFI ◽  
R. FAKHAR
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Iterative Method ◽  
Penalty Method ◽  
Original Problem ◽  
Unilateral Contact ◽  
Existence Of A Solution ◽  
Mathematical Properties ◽  
Element System ◽  
Nonlocal Friction

In this paper, we consider the penalty method to solve the unilateral contact with friction between an electro-elastic body and a conductive foundation. Mathematical properties, such as the existence of a solution to the penalty problem and its convergence to the solution of the original problem, are reported. Then, we present a finite elements approximation for the penalised problem and prove its convergence. Finally, we propose an iterative method to solve the resulting finite element system and establish its convergence.

scholarly journals A structure-exploiting numbering algorithm for finite elements on extruded meshes, and its performance evaluation in Firedrake

2016 ◽  
Vol 9 (10) ◽  
pp. 3803-3815 ◽  
Author(s):  
Gheorghe-Teodor Bercea ◽  
Andrew T. T. McRae ◽  
David A. Ham ◽  
Lawrence Mitchell ◽  
Florian Rathgeber ◽  
...  
Keyword(s):  
Finite Element ◽  
Performance Evaluation ◽  
Finite Elements ◽  
Aspect Ratio ◽  
Three Dimensional ◽  
Data Layout ◽  
Element System ◽  
Continuous Galerkin ◽  
Performance Penalty ◽  
The Cost

Abstract. We present a generic algorithm for numbering and then efficiently iterating over the data values attached to an extruded mesh. An extruded mesh is formed by replicating an existing mesh, assumed to be unstructured, to form layers of prismatic cells. Applications of extruded meshes include, but are not limited to, the representation of three-dimensional high aspect ratio domains employed by geophysical finite element simulations. These meshes are structured in the extruded direction. The algorithm presented here exploits this structure to avoid the performance penalty traditionally associated with unstructured meshes. We evaluate the implementation of this algorithm in the Firedrake finite element system on a range of low compute intensity operations which constitute worst cases for data layout performance exploration. The experiments show that having structure along the extruded direction enables the cost of the indirect data accesses to be amortized after 10–20 layers as long as the underlying mesh is well ordered. We characterize the resulting spatial and temporal reuse in a representative set of both continuous-Galerkin and discontinuous-Galerkin discretizations. On meshes with realistic numbers of layers the performance achieved is between 70 and 90 % of a theoretical hardware-specific limit.

scholarly journals A New Probabilistic Interpretation of the Bramble–Hilbert Lemma

2020 ◽  
Vol 20 (1) ◽  
pp. 79-87 ◽  
Author(s):  
Joël Chaskalovic ◽  
Franck Assous
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Probability Distributions ◽  
Mesh Size ◽  
Random Variable ◽  
The Other ◽  
Probabilistic Interpretation ◽  
Speed Of Convergence ◽  
Mathematical Properties ◽  
Asymptotic Speed

AbstractThe aim of this paper is to provide new perspectives on relative finite element accuracy which is usually based on the asymptotic speed of convergence comparison when the mesh size h goes to zero. Starting from a geometrical reading of the error estimate due to the Bramble–Hilbert lemma, we derive two probability distributions that estimate the relative accuracy, considered as a random variable, between two Lagrange finite elements {P_{k}} and {P_{m}} ({k<m}). We establish mathematical properties of these probabilistic distributions and we get new insights which, among others, show that {P_{k}} or {P_{m}} is more likely accurate than the other, depending on the value of the mesh size h.

scholarly journals A New Mixed Functional-probabilistic Approach for Finite Element Accuracy

2020 ◽  
Vol 20 (4) ◽  
pp. 799-813
Author(s):  
Joël Chaskalovic ◽  
Franck Assous
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Asymptotic Relation ◽  
Probabilistic Approach ◽  
Random Variable ◽  
High Order ◽  
Relative Accuracy ◽  
The Difference ◽  
New Perspective

AbstractThe aim of this paper is to provide a new perspective on finite element accuracy. Starting from a geometrical reading of the Bramble–Hilbert lemma, we recall the two probabilistic laws we got in previous works that estimate the relative accuracy, considered as a random variable, between two finite elements {P_{k}} and {P_{m}} ({k<m}). Then we analyze the asymptotic relation between these two probabilistic laws when the difference {m-k} goes to infinity. New insights which qualify the relative accuracy in the case of high order finite elements are also obtained.

The BERSAFE finite element system

1974 ◽  
Vol 6 (1) ◽  
pp. 15-24 ◽  
Author(s):  
T.K. Hellen ◽  
S.J. Protheroe
Keyword(s):  
Finite Element ◽  
Element System

DISCRETE COMPACTNESS FOR p AND hp 2D EDGE FINITE ELEMENTS

2003 ◽  
Vol 13 (11) ◽  
pp. 1673-1687 ◽  
Author(s):  
DANIELE BOFFI ◽  
LESZEK DEMKOWICZ ◽  
MARTIN COSTABEL
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Model Problem ◽  
Finite Element Approximation ◽  
Stability Estimate ◽  
Element Approximation ◽  
Dimensional Model ◽  
Compactness Property ◽  
One Dimensional ◽  
Discrete Compactness

In this paper we discuss the hp edge finite element approximation of the Maxwell cavity eigenproblem. We address the main arguments for the proof of the discrete compactness property. The proof is based on a conjectured L2 stability estimate for the involved polynomial spaces which has been verified numerically for p≤15 and illustrated with the corresponding one dimensional model problem.

Performance of a Stirling Engine Regenerator Having Finite Mass

1986 ◽  
Vol 108 (4) ◽  
pp. 669-673 ◽  
Author(s):  
J. D. Jones
Keyword(s):  
Finite Element ◽  
Pressure Variation ◽  
Stirling Engine ◽  
System Model ◽  
Finite Mass ◽  
Element System ◽  
The Matrix ◽  
Cyclic Variations ◽  
Good Agreement ◽  
Engine Power

The performance of a Stirling engine regenerator subjected to sinusoidal mass flow rate and pressure variation is analyzed. It is shown that cyclic variations in the temperature of the matrix due to its finite mass lead to an increase in the apparent regenerator effectiveness, but a decrease in engine power. Approximate closed-form expressions for both of these effects are deduced. The results of this analysis are compared with the predictions of a finite-element system model, and good agreement is found.

Penalty Method Involving Electric Sliding Contact Problem

2014 ◽  
Vol 701-702 ◽  
pp. 246-249
Author(s):  
Sai Tan ◽  
Jun Yong Lu ◽  
Xin Lin Long ◽  
Xiao Zhang
Keyword(s):  
Finite Element ◽  
Contact Problem ◽  
Physical Quantity ◽  
Penalty Method ◽  
Sliding Contact ◽  
Interface Condition ◽  
Calculation Results ◽  
Coupled Equation ◽  
Finite Element Formulations ◽  
Methods Numerical

Basing on governing Maxwell and energy equation of rail gun considering armature movement in two dimension, The total domain to be solved is divided into two subdomains: moving (armature) part and static (rail) part, finite element formulations of two subdomains are built independently, then using the interface condition of two subdomains, formulations are connected by coupled equation which is derived out by penalty method. Shifted physical quantity is used to simulate movement. The final magnetic-thermal coupled fields finite element formulations of rail gun are established by these methods. Numerical calculation results compared by theoretical and other numerical results verify that penalty method is an effective way to deal with electric sliding contact problem associating with Shifted physical quantity method.

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