Abstract

A stabilized finite element method (FEM) for the multidimensional steady state advection-diffusion-absorption equation is presented. The stabilized formulation is based on the modified governing differential equations derived via the Finite Calculus (FIC) method. For 1D problems the stabilization terms act as a nonlinear additional diffusion governed by a single stabilization parameter. It is shown that for multidimensional problems an orthotropic stabilizing diffusion must be added along the principal directions of  curvature of the solution. A simple iterative algorithm yielding a stable and accurate solution for all the range of physical parameters and boundary conditions is described. Numerical results for 1D and 2D problems with sharp gradients are presented showing the effectiveness and accuracy of the new stabilized formulation.

Abstract

Nowadays, fluid-structure interaction problems are a great challenge of different fields in engineering and applied sciences. In civil engineering applications, wind flow and structural motion may lead to aeroelastic instabilities on constructions such as long-span bridges, high-rise buildings and light-weight roof structures. On the other hand, biomechanical applications are interested in the study of hemodynamics, i.e. blood flow through large arteries, where large structural membrane deformations interact with incompressible fluids.   In the structural part of this work, a new methodology for the analysis  of geometrically nonlinear orthotropic membrane and rotation-free  shell elements is developed based on the principal fiber orientation  of the material. A direct consequence of the fiber orientation strategy is the possibility to analyze initially out-of-plane prestressed membrane and shell structures. Additionally, since conventional membrane theory  allows compression stresses, a wrinkling algorithm based on modifying the constitutive equation is presented. The structure is modeled with finite elements emerging from the governing equations of elastodynamics.   The fluid portion of this work is governed by the incompressible Navier-Stokes equations, which are modeled by stabilized equal-order interpolation finite elements. Since the monolithic solution for these equations has the disadvantage that take great computer effort to solve large algebraic system of equations, the fractional step methodology is used to take advantage of the computational efficiency given by the uncoupling of the pressure from the velocity field.  In addition, the generalized-a time integration scheme for fluids is  adapted to be used with the fractional step technique.   The fluid-structure interaction problem is formulated as a three-field system: the structure, the fluid and the moving fluid mesh solver. Motion of the fluid domain is accounted for with the arbitrary  Lagrangian-Eulerian formulation with two different mesh update strategies. The coupling between the fluid and the structure is performed with the strong coupling block Gauss-Seidel partitioned technique.  Since the fluid-structure interaction problem is highly nonlinear,  a relaxation technique based on Aitken's method is implemented in the strong coupling formulation to accelerate the convergence.   Finally several example problems are presented in each field to verify the robustness and efficiency of the overall algorithm,  many of them validated with reference solutions.