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Least-squares mixed finite elements for geometrically nonlinear solid mechanics
The computation of reliable results using finite elements is a major engineering goal. Under the assumption of a linear elastic theory many stable and reliable (standard and mixed) finite elements have been developed. Unfortunately, in the geometrically nonlinear regime, e.g. applying these elements in the field of incompressible, hyperelastic materials, problems can occur. A possible approach to circumvent these issues might be the least-squares mixed finite element method. Therefore, in this thesis, a mixed least-squares formulation for hyperelastic materials in the field of solid mechanics is provided, investigated and valuated. To create a theoretical basis the continuum mechanical background is outlined, the necessary physical quantities are introduced and the construction of suitable interpolation functionsis shown. Furthermore, the general procedure for the construction of a least-squares functional is described and applied for hyperelastic material laws based on a free energy function. Basis for the proposed least-squares element formulation is a div-grad first-order system consisting of the equilibrium condition, the constitutive equation and a stress symmetry condition, all written in a residual form. The solution variables (displacements and stresses) are, dependent on the element type, interpolated using different approximation spaces. The performance of the provided elements is investigated and compared to standard and mixed Galerkin elements by extensive numerical studies with respect to e.g. bending dominated problems, incompressibility, stability issues, convergence of the field quantities and adaptivity. Furthermore, the crucial influence of weighting is discussed. Finally, the results are evaluated and the used elements are assessed. ; Ein Hauptziel im Bereich des Ingenieurwesens ist die Berechnung vertrauenswürdiger Ergebnisse mit Hilfe der Methode der finiten Elemente. Unter Annahme einer linear elastischen Theorie wurden hierzu bereits viele stabile und zuverlässige standard und gemischte ...
Least-squares mixed finite elements for geometrically nonlinear solid mechanics
The computation of reliable results using finite elements is a major engineering goal. Under the assumption of a linear elastic theory many stable and reliable (standard and mixed) finite elements have been developed. Unfortunately, in the geometrically nonlinear regime, e.g. applying these elements in the field of incompressible, hyperelastic materials, problems can occur. A possible approach to circumvent these issues might be the least-squares mixed finite element method. Therefore, in this thesis, a mixed least-squares formulation for hyperelastic materials in the field of solid mechanics is provided, investigated and valuated. To create a theoretical basis the continuum mechanical background is outlined, the necessary physical quantities are introduced and the construction of suitable interpolation functionsis shown. Furthermore, the general procedure for the construction of a least-squares functional is described and applied for hyperelastic material laws based on a free energy function. Basis for the proposed least-squares element formulation is a div-grad first-order system consisting of the equilibrium condition, the constitutive equation and a stress symmetry condition, all written in a residual form. The solution variables (displacements and stresses) are, dependent on the element type, interpolated using different approximation spaces. The performance of the provided elements is investigated and compared to standard and mixed Galerkin elements by extensive numerical studies with respect to e.g. bending dominated problems, incompressibility, stability issues, convergence of the field quantities and adaptivity. Furthermore, the crucial influence of weighting is discussed. Finally, the results are evaluated and the used elements are assessed. ; Ein Hauptziel im Bereich des Ingenieurwesens ist die Berechnung vertrauenswürdiger Ergebnisse mit Hilfe der Methode der finiten Elemente. Unter Annahme einer linear elastischen Theorie wurden hierzu bereits viele stabile und zuverlässige standard und gemischte ...
Least-squares mixed finite elements for geometrically nonlinear solid mechanics
Steeger, Karl (author) / Schröder, Jörg
2017-07-13
Theses
Electronic Resource
English
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