Associated people

A. Schwarz, J. Schröder, S. Averweg und C. Nisters

Abstract

The proposed research project aims to develop, construct and analyse least-squares finite element methods (LSFEM) in fluid- and solid mechanics. The proposed mixed element formulations are given in stresses and velocities and are supposed to be used in a new method of fluid-structure interaction (FSI). Current FSI approaches include a certain treatment of the interface between solid and fluid phase in order to guarantee the equality of the mentioned quantities along this interface. In the proposed FSI approach these interface conditions are fulfilled intrinsically by the choice of the interpolation spaces and are considered automatically.The LSFEM results in stable formulations for first order systems and is applied successfully especially in fluid mechanics. The construction of the finite elements leads to positive definite and symmetric system matrices, also for differential equations with not self-adjoint operators. Furthermore, the method is not restricted to the LBB-condition, such that a free choice of the polynomial degree of the interpolating functions is possible. This increases the number of possible combinations in the construction of the finite elements.Essentially, the proposed research project can be subdivided into three periods. First, in a preinvestigation the central idea of the FSI coupling is analysed for a small strain model. This is the fundament for the following stage of constructing and developing the proposed mixed finite elements for the fluid and solid phase considering finite deformations and instationarities. The stress-velocity formulations are investigated with respect to reliability and efficiency. Special focus rests on the approximation quality of the solution variables and the robustness of the formulations. Finally, the least-squares FSI approach has to be validated with help of known benchmark problems. The research project is supposed to point the way in the regime of non-conventional discretization methods solving fluid-structure-interaction problems.

References

Nisters C, Schwarz A, Averweg S and Schröder J (2018), "Remarks on a
Fluid-Structure Interaction scheme based on the least-squares finite
element method at small strains", Advances in Mechanics of Materials and
Structural Analysis. Advanced Structured Materials 80, pp 261-279,
Springer.

Schwarz A, Nisters C, Averweg S and Schröder J (2018), "Stress-Velocitiy
Mixed Least-Squares FEMs for the Time-Dependent Incompressible
Navier-Stokes Equations", In Large-Scale Scientific Computing. Lecture
Notes in Computer Science. Vol. 10665, pp. 137-144. Springer.

Nisters C and Schwarz A (2018). "Efficient stress–velocity least-squares
finite element formulations for the incompressible Navier–Stokes equations".
Computer Methods in Applied Mechanics and Engineering, 341, 333–359.