A modified least‐squares mixed finite element with improved momentum balance

Igelbüscher

et al. 2016

The main goal of this contribution is the solution of geometrically nonlinear problems using the mixed least-squares finite element method (LSFEM). An investigation of a hyperelastic material law based on logarithmic deformation measures is performed. The basis for the proposed LSFEM is a div-grad first-order system consisting of the equilibrium condition and the constitutive equation, see e.g. Cai and Starke [1]. For the interpolation of the solution variables vector-valued Raviart-Thomas functions for the approximation of the stresses and standard Lagrange polynomials for the displacements are used. In order to show the performance of the presented formulations a numerical example is investigated, where we compare the different interpolation combinations used.

Section: Theoretical Frameworkmentioning

confidence: 99%

Comparison of different least‐squares mixed finite element formulations for hyperelasticity

Igelbüscher

et al. 2016

“…with m as the order of the standard polynomials (P ) interpolating the displacements and k denoting the approximation order for the stresses interpolated by vector-valued Raviart-Thomas (RT ) functions, leading to an element RT m P k , compare also [2].…”

Section: Theoretical Frameworkmentioning

confidence: 99%

Comparison of geometrically nonlinear LSFEM formulations based on different hyperelastic models

Schröder

et al. 2013

This contribution deals with the solution of geometrically nonlinear elastic problems solved by the least-squares mixed finite element method (LSFEM). The degrees of freedom (displacements and stresses) will be approximated using suitable spaces, namely W 1,p with p > 4 and H(div, Ω). In order to define the stress response of the material, different hyperelastic free energy functions will be presented. The residual forms R i of the balance of momentum and the constitutive equation build a system of differential equations of first order. Choosing suitable weighting operators and applying L 2 -norms lead to a leastsquares functional F (P ,u). The interpolation of the unknowns is accomplished using a standard polynomial interpolation for the displacements and vector-valued Raviart-Thomas functions for the approximation of the stresses. The formulations presented will be compared considering a uni-axial spatial tension test.

“…For the stresses vector-valued Raviart-Thomas interpolants are used and for the approximation of the displacements a standard polynomial interpolation is chosen, see also [1]. This leads to a finite element labeled by RT m P k , where m denotes the interpolation order of the stresses and k the interpolation order for the displacements.…”

Section: Standard and Weighted Least-squares Formulationmentioning

confidence: 99%

“…The L 2 -norm minimization of the time-discretized residuals of the given first-order system of partial differential equations leads to a functional depending on displacements and stresses. In the numerical example the proposed mixed element is compared to an alternative approach, which is based on a least-squares mixed finite element with improved momentum balance, see [1]. …”

mentioning

confidence: 99%

A Weighted Least‐Squares Mixed Finite Element Formulation for Quasi‐Incompressible Elastodynamics

Schröder

2011

The purpose of this work is the application of the least-squares finite element method to an elastodynamic, quasi-incompressible problem under small strain assumptions. Therefore a mixed finite element based on a weighted least-squares formulation is developed. The L 2 -norm minimization of the time-discretized residuals of the given first-order system of partial differential equations leads to a functional depending on displacements and stresses. In the numerical example the proposed mixed element is compared to an alternative approach, which is based on a least-squares mixed finite element with improved momentum balance, see [1].