scholarly journals Finite element estimates for a class of nonlinear variational inequalities

1993 ◽  
Vol 16 (3) ◽  
pp. 503-509 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Applied Sciences ◽  
Finite Element ◽  
Approximate Solution ◽  
Variational Inequalities ◽  
Error Estimates ◽  
Obstacle Problem ◽  
Nonlinear Problems ◽  
Highly Nonlinear ◽  
Unilateral Problems ◽  
Special Case

It is well known that a wide class of obstacle and unilateral problems arising in pure and applied sciences can be studied in a general and unifield framework of variational inequalities. In this paper, we derive the error estimates for the finite element approximate solution for a class of highly nonlinear variational inequalities encountered in the field of elasticity and glaciology in terms ofW1,p(Ω)andLp(Ω)-norms. As a special case, we obtain the well-known error estimates for the corresponding linear obstacle problem and nonlinear problems.

Optimization of the Closure-Weld Region of Cylindrical Containers for Long-Term Corrosion Resistance Using the Successive Heuristic Quadratic Approximation Technique*

2003 ◽  
Vol 125 (3) ◽  
pp. 533-539 ◽  
Author(s):  
Zekai Ceylan ◽  
Mohamed B. Trabia
Keyword(s):  
Finite Element ◽  
Random Search ◽  
Nonlinear Problems ◽  
Search Space ◽  
Induction Coil ◽  
Quadratic Approximation ◽  
Approximation Technique ◽  
General Corrosion ◽  
Highly Nonlinear

Welded cylindrical containers are susceptible to stress corrosion cracking (SCC) in the closure-weld area. An induction coil heating technique may be used to relieve the residual stresses in the closure-weld. This technique involves localized heating of the material by the surrounding coils. The material is then cooled to room temperature by quenching. A two-dimensional axisymmetric finite element model is developed to study the effects of induction coil heating and subsequent quenching. The finite element results are validated through an experimental test. The container design is tuned to maximize the compressive stress from the outer surface to a depth that is equal to the long-term general corrosion rate of the container material multiplied by the desired container lifetime. The problem is subject to several geometrical and stress constraints. Two different solution methods are implemented for this purpose. First, an off-the-shelf optimization software is used. The results however were unsatisfactory because of the highly nonlinear nature of the problem. The paper proposes a novel alternative: the Successive Heuristic Quadratic Approximation (SHQA) technique. This algorithm combines successive quadratic approximation with an adaptive random search within varying search space. SHQA promises to be a suitable search method for computationally intensive, highly nonlinear problems.

A Consistent Implicit Formulation for Nonlinear Finite Element Modeling With Contact and Friction: Part I—Theory

1991 ◽  
Vol 58 (2) ◽  
pp. 499-506 ◽  
Author(s):  
M. J. Saran ◽  
R. H. Wagoner
Keyword(s):  
Finite Element ◽  
Nonlinear Problems ◽  
Material Model ◽  
Test Problems ◽  
Large Strains ◽  
Numerical Verification ◽  
Viscoplastic Material ◽  
Highly Nonlinear ◽  
Complex Boundary ◽  
Consistent Tangent Operator

A formulation for finite element simulation of highly nonlinear problems including friction and contact with arbitrarily shaped rigid surfaces is proposed (CFS approach), prompted by difficulties in robust and accurate simulations of industrial forming processes. Nonlinearities are caused by large strains, plastic flow, and complex boundary conditions with frictional contact. In Part I the theoretical basis is described and the appropriate numerical algorithm is derived. The complete set of the governing relations, comprising equilibrium and interfacial equations, is appropriately linearized; resulting in a consistent tangent operator of the Newton-Raphson algorithm. In Part II, as a numerical verification, plane-strain sheet-forming processes are analyzed using a rigid-viscoplastic material model. Results are presented and discussed for test problems and for complex simulation of reverse drawing by concave tools.

scholarly journals Variational inequalities for energy functionals with nonstandard growth conditions

1998 ◽  
Vol 3 (1-2) ◽  
pp. 41-64 ◽  
Author(s):  
Martin Fuchs ◽  
Li Gongbao
Keyword(s):  
Variational Inequalities ◽  
Obstacle Problem ◽  
Partial Regularity ◽  
Growth Conditions ◽  
Energy Functionals ◽  
Nonstandard Growth ◽  
Bounded Lipschitz Domain ◽  
Growth Properties ◽  
Nonstandard Growth Conditions ◽  
Special Case

We consider the obstacle problem{minimize????????I(u)=?OG(?u)dx??among functions??u:O?Rsuch?that???????u|?O=0??and??u=F??a.e.for a given functionF?C2(O¯),F|?O<0and a bounded Lipschitz domainOinRn. The growth properties of the convex integrandGare described in terms of aN-functionA:[0,8)?[0,8)withlimt?8¯A(t)t-2<8. Ifn=3, we prove, under certain assumptions onG,C1,8-partial regularity for the solution to the above obstacle problem. For the special case whereA(t)=tln(1+t)we obtainC1,a-partial regularity whenn=4. One of the main features of the paper is that we do not require any power growth ofG.

Post-Buckling Analysis of Axially Functionally Graded Three-Dimensional Beams

2015 ◽  
Vol 07 (03) ◽  
pp. 1550047 ◽  
Author(s):  
Şeref Doğuşcan Akbaş
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Element Model ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Highly Nonlinear ◽  
Lagrangian Finite Element ◽  
Post Buckling

Post-buckling analysis of an axially functionally graded (AFG) cantilever beam subjected to an axial nonfollower compression load is studied in this paper by using the total Lagrangian finite element model of three-dimensional continuum approximations. Material properties of the beam change in the axial direction according to a power-law function. In this study, finite element model of the beam is constructed by using total Lagrangian finite element model of three-dimensional continuum for an eight-node quadratic element. It is known that post-buckling problems are geometrically nonlinear problems. The considered highly nonlinear problem is solved by using incremental displacement-based finite element method in conjunction with Newton–Raphson iteration method. There is no restriction on the magnitudes of deflections and rotations in contradistinction to von-Karman strain displacement relations. The obtained results are compared with the published results. In this study, the effects of the material distribution on the post-buckling response of the AFG beam are investigated in detail. The differences between of material distributions are investigated in the post-buckling analysis. Numerical results show that the above-mentioned effects play a very important role on the post-buckling responses of the beam, and it is believed that new results are presented for post-buckling of AFG beams which are of interest to the scientific and engineering community in the area of FGM structures.

scholarly journals Bubbles Enriched Quadratic Finite Element Method for the 3D-Elliptic Obstacle Problem

2018 ◽  
Vol 18 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Sharat Gaddam ◽  
Thirupathi Gudi
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Estimates ◽  
Obstacle Problem ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Three Dimensional ◽  
A Priori Error Estimates ◽  
Priori Error Estimates ◽  
Element Method

AbstractAn optimally convergent (with respect to the regularity) quadratic finite element method for the two-dimensional obstacle problem on simplicial meshes is studied in [14]. There was no analogue of a quadratic finite element method on tetrahedron meshes for the three-dimensional obstacle problem. In this article, a quadratic finite element enriched with element-wise bubble functions is proposed for the three-dimensional elliptic obstacle problem. A priori error estimates are derived to show the optimal convergence of the method with respect to the regularity. Further, a posteriori error estimates are derived to design an adaptive mesh refinement algorithm. A numerical experiment illustrating the theoretical result on a priori error estimates is presented.

Finite Element Analysis of the Rubber of the Prosthetic Knee Joint

2014 ◽  
Vol 607 ◽  
pp. 346-349
Author(s):  
Ying Liu ◽  
Xiu Feng Zhang ◽  
Yan Ma
Keyword(s):  
Finite Element ◽  
Nonlinear Problems ◽  
Element Model ◽  
Element Analysis ◽  
Rubber Material ◽  
Knee Motion ◽  
Prosthetic Knee ◽  
Rubber Bearing ◽  
Highly Nonlinear ◽  
Selection Of

Rubber played a buffer role in prosthetic knee motion. Rubber bearing is a very complicated process, and rubber material itself is nonlinear. ABAQUS software can able to deal with highly nonlinear problems. Input rubber test data in ABAQUS, selection of constitutive model, and then the finite element model is established, which is calculated, finally obtains compression under different loads.

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