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Mixed least squares finite element methods based on inverse stress-strain relations in hyperelasticity
Reliable simulation techniques for the description of elastic deformation processes in solid mechanics are nowadays of great importance. A reasonable model should take nonlinear kinematics and a nonlinear material law into account and should coincide with Hooke’s law under small loads. In addition, a numerical method should be able to simulate compressible as well as (almost) incompressible material behavior. The calculation of good stress and displacement approximations is often of particular interest. Therefore general mixed least squares finite element methods in the context of finite hyperelasticity are considered in this work. They are based on the conservation of linear momentum and inverse stress-strain relations and will be used for the simulation of homogeneous isotropic and homogeneous transverse isotropic material behavior. For the minimization of the nonlinear least squares functionals in finite dimensional spaces a Gauss-Newton framework is applied. In the case of a specific homogeneous isotropic Neo-Hooke model an analysis is provided which proves reliability and efficiency of the nonlinear least squares functional as a-posteriori error estimator. The analysis remains valid in the incompressible limit and therefore the Poisson locking effect is excluded. The analytical results for the Neo-Hooke model are used to propose an algorithm for model adaptivity which is based on the model of linear elasticity and the Neo-Hooke model. The algorithm automatically decides in which subdomain the linear model should be locally substituted by the Neo-Hooke model. Two- and three-dimensional numerical examples for compressible and fully incompressible materials are given in order to illustrate the potential of our method. Here next-to-lowest-order Raviart-Thomas elements for the stress approximations are combined with conforming piecewise quadratic elements for the displacement approximations. A significant improvement of stress approximations in comparison to conventional discretization methods is demonstrated. ...
Mixed least squares finite element methods based on inverse stress-strain relations in hyperelasticity
Reliable simulation techniques for the description of elastic deformation processes in solid mechanics are nowadays of great importance. A reasonable model should take nonlinear kinematics and a nonlinear material law into account and should coincide with Hooke’s law under small loads. In addition, a numerical method should be able to simulate compressible as well as (almost) incompressible material behavior. The calculation of good stress and displacement approximations is often of particular interest. Therefore general mixed least squares finite element methods in the context of finite hyperelasticity are considered in this work. They are based on the conservation of linear momentum and inverse stress-strain relations and will be used for the simulation of homogeneous isotropic and homogeneous transverse isotropic material behavior. For the minimization of the nonlinear least squares functionals in finite dimensional spaces a Gauss-Newton framework is applied. In the case of a specific homogeneous isotropic Neo-Hooke model an analysis is provided which proves reliability and efficiency of the nonlinear least squares functional as a-posteriori error estimator. The analysis remains valid in the incompressible limit and therefore the Poisson locking effect is excluded. The analytical results for the Neo-Hooke model are used to propose an algorithm for model adaptivity which is based on the model of linear elasticity and the Neo-Hooke model. The algorithm automatically decides in which subdomain the linear model should be locally substituted by the Neo-Hooke model. Two- and three-dimensional numerical examples for compressible and fully incompressible materials are given in order to illustrate the potential of our method. Here next-to-lowest-order Raviart-Thomas elements for the stress approximations are combined with conforming piecewise quadratic elements for the displacement approximations. A significant improvement of stress approximations in comparison to conventional discretization methods is demonstrated. ...
Mixed least squares finite element methods based on inverse stress-strain relations in hyperelasticity
Müller, Benjamin (author) / Starke, Gerhard
2015-05-22
Theses
Electronic Resource
English
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