Numerical methods for static shallow shells lying over an obstacle
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Numerical methods for static shallow shells lying over an obstacle Paolo Piersanti1
· Xiaoqin Shen2
Received: 26 September 2019 / Accepted: 4 October 2019 / © The Author(s) 2020
Abstract In this paper, a finite element analysis to approximate the solution of an obstacle problem for a static shallow shell confined in a half space is presented. To begin with, we establish, by relying on the properties of enriching operators, an estimate for the approximate bilinear form associated with the problem under consideration. Then, we conduct an error analysis and we prove the convergence of the proposed numerical scheme. Keywords Shallow shell · Enriching operator · Nonconforming finite element method · Obstacle problems · Elliptic variational inequalities
1 Introduction The study of unilateral contact problems in elasticity arises in many applicative fields such as structural mechanics and civil engineering. Obstacle problems have lately been studied in, for instance, [20, 21, 23, 24, 36, 40]. The numerical analysis of obstacle problems has been arising the interest of many scientists since the late 1990s. In this direction, a very direct and mathematically elegant approach is the one making use of enriching operators, the properties of which were studied by S. C. Brenner and her collaborators in the seminal papers [1–3, 5]. These general theoretical results were then used to study finite element methods for
Paolo Piersanti
[email protected] Xiaoqin Shen [email protected] 1
Institute of Mathematics and Scientific Computing, Karl-Franzens-Universit¨at Graz, Heinrichstraße 36, A8010, Graz, Austria
2
Department of Mathematics, School of Sciences, Xi’an University of Technology, P.O. Box 1243, Yanxiang Road No. 58, Xi’an, 710054, Shaanxi Province, China
Numerical Algorithms
obstacle problems, which can be found in [7] and [6]. Nonconforming finite element methods for obstacle problem were also studied in [11]. In this paper, we study the displacement of a static shallow shell lying over a planar obstacle from the numerical point of view, using a suitable finite element method. Shallow shells theory, which is extensively described, for instance, in the books [16] and [44], is widely used in engineering (see, e.g., the papers [31, 41–43, 46]). According to this theory, the problem under examination is modelled in terms of a fourth-order differential operator (cf., e.g., [16]). The theory of finite element methods for fourth-order problems governed by variational inequalities has been investigated, for instance, in the references [8, 25, 30, 32]. Our mathematical model of an obstacle problem for a linearly elastic shallow shell in the static case is inspired by that of L´eger and Miara (cf. [34] and [35]). To our best knowledge, there is no reference on the study of numerical analysis of obstacle problems for linearly elastic shallow shells. In this paper, we extend the method proposed in [7] and [6] to derive error estimates for the solution to the obstacle problem under for linearly elastic shallow shel
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