73 research outputs found
Error estimations of mixed finite element methods for nonlinear problems of shallow shell theory
© Published under licence by IOP Publishing Ltd.The variational formulations of problems of equilibrium of a shallow shell in the framework of the geometrically and physically nonlinear theory by boundary conditions of different main types, including non-classical, are considered. Necessary and sufficient conditions for their solvability are derived. Mixed finite element methods for the approximate solutions to these problems based on the use of second derivatives of the bending as auxiliary variables are proposed. Estimations of accuracy of approximate solutions are established
On a class of grid approximations for nonlinear problems in plate theory
Mixed schemes with numerical integration are suggested for geometrically and physically nonlinear problems in plate theory. Solvability conditions are obtained, and the accuracy of the schemes is estimated. Copyright © 1999 by MAHK "Hayxa/Interperiodica"
Sharp interface limit for a phase field model in structural optimization
We formulate a general shape and topology optimization problem in structural
optimization by using a phase field approach. This problem is considered in
view of well-posedness and we derive optimality conditions. We relate the
diffuse interface problem to a perimeter penalized sharp interface shape
optimization problem in the sense of -convergence of the reduced
objective functional. Additionally, convergence of the equations of the first
variation can be shown. The limit equations can also be derived directly from
the problem in the sharp interface setting. Numerical computations demonstrate
that the approach can be applied for complex structural optimization problems
An iterative method for mixed finite element schemes
An iterative method with a saddle preconditioner is proposed for solving a system of nonlinear equations that arises in the approximation of a quasilinear second-order elliptic equation with a mixed scheme of finite elements of Raviart-Thomas type. The ways of choosing the iteration parameter are pointed out that ensure the convergence of the method. The results of numerical experiments are presented. © 2012 Pleiades Publishing, Ltd
Numerical solution of an inverse problem of determining the parameters of a source of groundwater pollution
The article deals with an inverse problem of determining parameters of groundwater pollution sources. We test three ways to solve the problem on the simulated data for a simple case of contamination. We discover that, in the presence of noise in the data of the inverse problem, the first method does not produce satisfactory recovery results, while the second and third ones are comparable in accuracy of the recovery of required parameters. Taking into account the ease of implementation, the speed of computing and parallelization feasibility the second method of solving the inverse problem is found to be most preferable. We also propose a method of finding pollution parameters in general case..
Mixed schemes of finite element method for non-standard boundary value problems of the nonlinear theory of thin elastic shells
© 2019 IOP Publishing Ltd. Variational statements of equilibrium problems for a thin elastic shell within a geometrically nonlinear theory of mean bending for different types of principal boundary conditions, including non-classical, are given. Two classes of shell models are considered: (1) a geometrically nonlinear equilibrium model for an anisotropic shell of a material obeying generalized Hooke's law; (2) a geometrically and physically nonlinear equilibria model for a shallow shell. Sufficient conditions for their generalized solvability in the corresponding Sobolev spaces are derived. In the case of the non-shallow shell the implicit function theorem is used. In the case of the shallow shell the variation problem is investigated using the generalized Weierstrass principle. Mixed finite element methods based on the use of the second derivatives of the deflection as auxiliary variables are constructed for approximate solutions of these problems. Sufficient conditions for solvability of the corresponding discrete problems are obtained. The convergence of approximate solutions is investigated. Accuracy estimates in the case of sufficiently smooth solutions of the original problems are given. Additional conditions are proposed for the problem on the shallow shell to ensure of implementation of the inequalities of the type of strong monotonicity and Lipschitz-continuity of the differential operator
Lagrangian Mixed Finite Element Methods for Nonlinear Thin Shell Problems
© 2019 Walter de Gruyter GmbH, Berlin/Boston 2019. A class of Lagrangian mixed finite element methods is constructed for an approximate solution of a problem of nonlinear thin elastic shell theory, namely, the problem of finding critical points of the functional of potential energy according to the Budiansky-Sanders model. The proposed numerical method is based on the use of the second derivatives of the deflection as auxiliary variables. Sufficient conditions for the solvability of the corresponding discrete problem are obtained. Accuracy estimates for approximate solutions are established. Iterative methods for solving the corresponding systems of nonlinear equations are proposed and investigated
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