SUMMARYThe paper presents a semi-implicit algorithm for solving an unsteady fluid-structure interaction problem. The algorithm for solving numerically the fluid-structure interaction problems was obtained by combining the backward Euler scheme with a semi-implicit treatment of the convection term for the Navier-Stokes equations and an implicit centered scheme for the structure equations. The structure is governed either by the linear elasticity or by the non-linear St Venant-Kirchhoff elasticity models. At each time step, the position of the interface is predicted in an explicit way. Then, an optimization problem must be solved, such that the continuity of the velocity as well as the continuity of the stress hold at the interface. During the Broyden, Fletcher, Goldforb, Shano (BFGS) iterations for solving the optimization problem, the fluid mesh does not move, which reduces the computational effort. The term 'semi-implicit' used for the fully algorithm means that the interface position is computed explicitly, while the displacement of the structure, velocity and the pressure of the fluid are computed implicitly. Numerical results are presented.
SUMMARYThe unsteady interaction between an incompressible fluid and a deformable elastic structure is analyzed. An implicit numerical method is proposed. At each time step, the stresses at the fluid-structure interface are determined as a solution of an optimization problem. The modal decomposition of the structure equations leads to a problem to be solved with a reduced number of unknowns. The analytic gradient of the cost function was derived. Numerical tests validate the analytic derivative and show the behavior of a two-dimensional Navier-Stokes equations with plate-like model interaction.
We describe finite element simulations of limb growth based on Stokes flow models with a nonzero divergence representing growth due to nutrients in the early stages of limb bud development. We introduce a "tissue pressure" whose spatial derivatives yield the growth velocity in the limb and our explicit time advancing algorithm for such tissue flows is described in de tail. The limb boundary is approached by spline functions to compute the curvature and the unit outward normal vector. At each time step, a mixed hybrid finite element problem is solved, where the condition that the velocity is strictly normal to the limb boundary is treated by a Lagrange multiplier technique. Numerical results are presented.
Abstract. An algorithm for approximation of an unsteady fluid-structure interaction problem is proposed. The fluid is governed by the Navier-Stokes equations with boundary conditions on pressure, while for the structure a particular plate model is used. The algorithm is based on the modal decomposition and the Newmark Method for the structure and on the Arbitrary Lagrangian Eulerian coordinates and the Finite Element Method for the fluid. In this paper, the continuity of the stresses at the interface was treated by the Least Squares Method. At each time step we have to solve an optimization problem which permits us to use moderate time step. This is the main advantage of this approach. In order to solve the optimization problem, we have employed the Broyden, Fletcher, Goldforb, Shano Method where the gradient of the cost function was approached by the Finite Difference Method. Numerical results are presented.Mathematics Subject Classification. 74F10, 75D05, 65M60.
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