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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Topology optimization of systems governed by variational inequalities

100%
Myśliński A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2010
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tom 30
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nr 2
237-252
EN
This paper deals with the formulation of the necessary optimality condition for a topology optimization problem of an elastic body in unilateral contact with a rigid foundation. In the contact problem of Tresca, a given friction is governed by an elliptic variational inequality of the second order. The optimization problem consists in finding such topology of the domain occupied by the body that the normal contact stress along the contact boundary of the body is minimized. The topological derivative of the cost functional is calculated and a necessary optimality condition is formulated. The calculated topological derivative is also used in the numerical algorithm to find a descent direction by inserting voids in the domain occupied by the body. Numerical examples are provided.
2
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Numerical considerations of a hybrid proximal projection algorithm for solving variational inequalities

80%
Jager C.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2007
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tom 27
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nr 1
51-69
EN
In this paper, some ideas for the numerical realization of the hybrid proximal projection algorithm from Solodov and Svaiter [22] are presented. An example is given which shows that this hybrid algorithm does not generate a Fejér-monotone sequence. Further, a strategy is suggested for the computation of inexact solutions of the auxiliary problems with a certain tolerance. For that purpose, ε-subdifferentials of the auxiliary functions and the bundle trust region method from Schramm and Zowe [20] are used. Finally, some numerical results for non-smooth convex optimization problems are given which compare the hybrid algorithm to the inexact proximal point method from Rockafellar [17].
3
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On the solution of a finite element approximation of a linear obstacle plate problem

80%
Fernandes L. M., Figueiredo I. N., Júdice J. J.
International Journal of Applied Mathematics and Computer Science
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2002
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tom 12
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nr 1
27-40
EN
In this paper the solution of a finite element approximation of a linear obstacle plate problem is investigated. A simple version of an interior point method and a block pivoting algorithm have been proposed for the solution of this problem. Special purpose implementations of these procedures are included and have been used in the solution of a set of test problems. The results of these experiences indicate that these procedures are quite efficient to deal with these instances and compare favourably with the path-following PATH and the active-set MINOS codes of the commercial GAMS collection.
4
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Topological derivatives for semilinear elliptic equations

70%
Iguernane M., Nazarov S., Roche J.-R., Sokolowski J., Szulc K.
International Journal of Applied Mathematics and Computer Science
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2009
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tom 19
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nr 2
191-205
EN
The form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the L∞ norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.
5
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A level set method in shape and topology optimization for variational inequalities

70%
Fulmański P., Laurain A., Scheid J.-F., Sokołowski J.
International Journal of Applied Mathematics and Computer Science
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2007
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tom 17
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nr 3
413-430
EN
The level set method is used for shape optimization of the energy functional for the Signorini problem. The boundary variations technique is used in order to derive the shape gradients of the energy functional. The conical differentiability of solutions with respect to the boundary variations is exploited. The topology modifications during the optimization process are identified by means of an asymptotic analysis. The topological derivatives of the energy shape functional are employed for the topology variations in the form of small holes. The derivation of topological derivatives is performed within the framework proposed in (Sokołowski and Żochowski, 2003). Numerical results confirm that the method is efficient and gives better results compared with the classical shape optimization techniques.
6
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Analysis of a frictionless contact problem for elastic bodies

70%
Drabla S., Sofonea M., Teniou B.
Annales Polonici Mathematici
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1998
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tom 69
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nr 1
75-88
EN
This paper deals with a nonlinear problem modelling the contact between an elastic body and a rigid foundation. The elastic constitutive law is assumed to be nonlinear and the contact is modelled by the well-known Signorini conditions. Two weak formulations of the model are presented and existence and uniqueness results are established using classical arguments of elliptic variational inequalities. Some equivalence results are presented and a strong convergence result involving a penalized problem is also proved.
7
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On discontinuous quasi-variational inequalities

61%
Chu L.-J., Lin C.-Y.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2007
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tom 27
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nr 2
199-212
EN
In this paper, we derive a general theorem concerning the quasi-variational inequality problem: find x̅ ∈ C and y̅ ∈ T(x̅) such that x̅ ∈ S(x̅) and ⟨y̅,z-x̅⟩ ≥ 0, ∀ z ∈ S(x̅), where C,D are two closed convex subsets of a normed linear space X with dual X*, and $T:X → 2^{X*}$ and $S:C → 2^D$ are multifunctions. In fact, we extend the above to an existence result proposed by Ricceri [12] for the case where the multifunction T is required only to satisfy some general assumption without any continuity. Under a kind of Karmardian's condition, we give a partial affirmative answer to an unbounded quasi-variational inequality problem.
8
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Analysis and numerical approximation of an elastic frictional contact problem with normal compliance

61%
Han W., Sofonea M.
Applicationes Mathematicae
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1999
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tom 26
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nr 4
415-435
EN
We consider the problem of frictional contact between an elastic body and an obstacle. The elastic constitutive law is assumed to be nonlinear. The contact is modeled with normal compliance and the associated version of Coulomb's law of dry friction. We present two alternative yet equivalent weak formulations of the problem, and establish existence and uniqueness results for both formulations using arguments of elliptic variational inequalities and fixed point theory. Moreover, we show the continuous dependence of the solution on the contact conditions. We also study the finite element approximations of the problem and derive error estimates. Finally, we introduce an iterative method to solve the resulting finite element system.
9
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A sign preserving mixed finite element approximation for contact problems

61%
Hild P.
International Journal of Applied Mathematics and Computer Science
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2011
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tom 21
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nr 3
487-498
EN
This paper is concerned with the frictionless unilateral contact problem (i.e., a Signorini problem with the elasticity operator). We consider a mixed finite element method in which the unknowns are the displacement field and the contact pressure. The particularity of the method is that it furnishes a normal displacement field and a contact pressure satisfying the sign conditions of the continuous problem. The a priori error analysis of the method is closely linked with the study of a specific positivity preserving operator of averaging type which differs from the one of Chen and Nochetto. We show that this method is convergent and satisfies the same a priori error estimates as the standard approach in which the approximated contact pressure satisfies only a weak sign condition. Finally we perform some computations to illustrate and compare the sign preserving method with the standard approach.
10
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Variational inequalities in noncompact nonconvex regions

51%
Lin C.-Y., Chu L.-J.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2003
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tom 23
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nr 1
5-19
EN
In this paper, a general existence theorem on the generalized variational inequality problem GVI(T,C,ϕ) is derived by using our new versions of Nikaidô's coincidence theorem, for the case where the region C is noncompact and nonconvex, but merely is a nearly convex set. Equipped with a kind of V₀-Karamardian condition, this general existence theorem contains some existing ones as special cases. Based on a Saigal condition, we also modify the main theorem to obtain another existence theorem on GVI(T,C,ϕ), which generalizes a result of Fang and Peterson.
11
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Numerical analysis and simulations of quasistatic frictionless contact problems

51%
García J. R. F., Han W., Shillor M., Sofonea M.
International Journal of Applied Mathematics and Computer Science
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2001
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tom 11
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nr 1
205-222
EN
A summary of recent results concerning the modelling as well as the variational and numerical analysis of frictionless contact problems for viscoplastic materials are presented. The contact is modelled with the Signorini or normal compliance conditions. Error estimates for the fully discrete numerical scheme are described, and numerical simulations based on these schemes are reported.
12
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New versions on Nikaidô's coincidence theorem

41%
Chu L.-J., Lin C.-Y.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2002
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tom 22
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nr 1
79-95
EN
In 1959, Nikaidô established a remarkable coincidence theorem in a compact Hausdorff topological space, to generalize and to give a unified treatment to the results of Gale regarding the existence of economic equilibrium and the theorems in game problems. The main purpose of the present paper is to deduce several generalized key results based on this very powerful result, together with some KKM property. Indeed, we shall simplify and reformulate a few coincidence theorems on acyclic multifunctions, as well as some Górniewicz-type fixed point theorems. Beyond the realm of monotonicity nor metrizability, the results derived here generalize and unify various earlier ones from the classic optimization theory. In the sequel, we shall deduce two versions of Nikaidô's coincidence theorem about Vietoris maps from different approaches.
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