Optimal Design of an Elastic Circular Plate on a Unilateral Elastic Foundation. I: Continuous Problems

Salač, Petr

doi:10.1002/1521-4001(200201)82:1<21::aid-zamm21>3.0.co;2-e

Cited by 3 publications

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“…(for the proof -see Salač [9] -Lemma 4.4). We observe that y ∈ V u , so that by (i) there exists a sequence {y n } such that y n ∈ Z u and y n → y in H 1 (Ω, Θ r ).…”

Section: Wwwzamm-journalorgmentioning

confidence: 99%

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Optimization of a functionally graded circular plate with inner rigid thin obstacles. I. Continuous problems

Hlaváček

Lovíšek

2011

Z Angew Math Mech

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Optimal control problems are considered for a functionally graded circular plate with inner rigid obstacles. Axisymmetric bending and stretching of the plate is studied using the classical Kirchhoff theory. The plate material is assumed to vary according to a power‐law distribution in terms of the volume fractions of the constituents. Four optimal design problems are considered for the elastic circular plate. The state problem is represented by a variational inequality with a monotone operator and the design variables (i.e., the thickness and the exponent of the power‐law) influence both the coefficients and the set of admissible state functions. We prove the existence of a solution to the above‐mentioned optimal design problems.

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“…(for the proof -see Salač [9] -Lemma 4.4). We observe that y ∈ V u , so that by (i) there exists a sequence {y n } such that y n ∈ Z u and y n → y in H 1 (Ω, Θ r ).…”

Section: Wwwzamm-journalorgmentioning

confidence: 99%

“…Since the embedding of H 2 (Ω, ρ r ) into C(Ω) is compact (see [9] -Lemma 4.5), w n → w in C(Ω). Passing to the limit in the conditions…”

Section: Wwwzamm-journalorgmentioning

confidence: 99%

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Optimization of a functionally graded circular plate with inner rigid thin obstacles. I. Continuous problems

Hlaváček

Lovíšek

2011

Z Angew Math Mech

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“…Recently the problem of fitting of thin plates solved in [3,4,5,6,7] finds wide use in automatic production of jumbo flat glass, in car industry, in architecture and in other fields. The aim is to minimize stress on plate such that it is not damaged or even destruction according to its specific material properties.…”

Section: Introductionmentioning

confidence: 99%

Problem of Determination of Optimal Supports Position for Jumbo Flat Glass with Free Edges

Salač

Horák

2008

AMR

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Recent trends in architecture are characterized with wide applications of laminated glass plates in facing systems of high-rise buildings, where the plates are oriented in both vertical and horizontal planes. This paper deals with mathematical description of optimization problem for location of finite number of rigid supports horizontally placed on jumbo flat glass plate with free edges. Associated state problem corresponding to Kirchhoff’s hypothesis of plate theory is solved. Further the optimization problem for finding location of the finite number of rigid supports, with respect to the condition for minimal deflection of the plate in the sense of 1 C -norm, is solved. Theoretical results are used in numerical simulation by the finite element method (FEM). It allows us to reduce the computation time for solution of real projects. Further application of such results is for example in the area of automatic handling in the flat glass production in the case of engineering design for location of vacuum gripping heads components of handling devices.

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Optimization of a functionally graded circular plate with inner rigid thin obstacles. II. Approximate problems

Hlaváček

Lovíšek

2011

Z Angew Math Mech

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Key wordsControl of elliptic variational inequalities, functionally graded plates, optimal design of plates, finite element approximations.Optimal design of a simply supported functionally graded axisymmetric circular plate resting on several inner rigid rings is presented in Part I. The variable thickness and the exponent of the power-law of the grading function are to be optimized.In Part II the approximate state problem and approximate optimal design problems are introduced, using spaces of linear and cubic Hermite splines, respectively. We prove the existence of approximate solutions and the convergence of a subsequence of the solutions to a solution of the original continuous optimal design problem.In Part II of the present paper we propose and analyse numerical approximate solutions of the four Optimal Design Problems introduced in Part I [2]. The procedure is based on simple finite element approximations of both the state and design variables.The core of the numerical analysis is represented by a general Theorem 3.1 on two-parametric approximations of elliptic variational inequalities. In Sect. 2 we introduce approximations of the set of admissible state variables, of the elliptic operators and of the designed thickness by means of linear and cubic Hermite splines.A general theorem on two-parametric approximations is proved in Sect. 3 and applied in Sect. 4. Approximate cost functionals are introduced in Sect. 5. In Sect. 6 we prove that there exists a subsequence of approximate optimal designs, which converges to a solution of the original Optimal Design Problem, if the mesh-size tends to zero. Approximate solution of the state problemWe introduce simple discretizations of the spaces, sets, and operators defined in Part I - [2].x m ] be a partition of the interval Ω ≡ [0, R], (x 0 = 0, x M = R), such that the points r 1 , . . . , r N , corresponding to the radii of rigid circular supports, belong to the nodes x m of T

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Optimal Design of an Elastic Circular Plate on a Unilateral Elastic Foundation. I: Continuous Problems

Cited by 3 publications

References 0 publications

Optimization of a functionally graded circular plate with inner rigid thin obstacles. I. Continuous problems

Optimization of a functionally graded circular plate with inner rigid thin obstacles. I. Continuous problems

Problem of Determination of Optimal Supports Position for Jumbo Flat Glass with Free Edges

Optimization of a functionally graded circular plate with inner rigid thin obstacles. II. Approximate problems

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