Implicit time discretization schemes for mixed least-squares finite element formulations

• Published in: Computer methods in applied mechanics and engineering Vol. 368; p. 113111
• Main Authors: Averweg, Solveigh, Schwarz, Alexander, Nisters, Carina, Schröder, Jörg
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.08.2020; Elsevier BV
• Subjects: Computational fluid dynamics; Differential equations; Discretization; Elastodynamics; Finite element method; Fluid flow; Fluid–structure interaction; Implicit time discretization schemes; Incompressible flow; Incompressible Navier–Stokes equations; Investigations; Least squares; Mathematical models; Mixed least-squares finite elements; Runge-Kutta method; Solid mechanics; Time integration; Fluid–structure interaction; Incompressible Navier–Stokes equations; Implicit time discretization schemes; Elastodynamics; Mixed least-squares finite elements
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2020.113111

This work is an extension of the ideas in Averweg et al. (2019) with the focus on a detailed investigation of implicit time discretization schemes to model instationary fluid flow, based on the incompressible Navier–Stokes equations, and linear elastodynamic structural behavior. The variational approaches for fluid and solid mechanics are based on a mixed least-squares finite element method. The L2-norm minimization of the residuals of the constructed first-order systems of the governing differential equations is based on two-field stress–velocity (SV) functionals. For the time discretization of the SV-fluid formulation, four different types of implicit integration schemes are investigated, namely the Houbolt method, the Crank–Nicolson method and two explicit, singly diagonally implicit Runge–Kutta methods (ESDIRK). The SV-formulation for the solid is discretized applying the Houbolt method. The presented time integration schemes are validated investigating an unsteady fluid flow and an elastodynamic structural benchmark. Since both (fluid and solid) SV formulations are discretized using conforming finite element spaces in H(div) and H1, respectively, the inherent fulfillment of coupling conditions, when modeling fluid–structure interaction problems, is given a priori. Therefore, the applicability is also examined by two simplified FSI problems for small deformations, in order to represent the main characteristics of the presented approach. •Investigation of different implicit time discretizations for mixed least-squares FEM.•Validation and verification for fluid and structure problems.•Accuracy assessment with respect to displacements and lift/drag coefficients.•Application to benchmark problems for FSI problems under small deformations.