Distributed Lagrange Multiplier/Fictitious Domain Finite Element Method for a Transient Stokes Interface Problem with Jump Coefficients

The distributed Lagrange multiplier/fictitious domain (DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients. The semi- and fully discrete DLM/FD-mixed finite element scheme are developed for the first time for this problem with a moving interface, where the arbitrary Lagrangian-Eulerian (ALE) technique is employed to deal with the moving and immersed subdomain. Stability and optimal convergence properties are obtained for both schemes. Numerical experiments are carried out for different scenarios of jump coefficients, and all theoretical results are validated

Finite Element Analysis of an Arbitrary Lagrangian–Eulerian Method for Stokes/Parabolic Moving Interface Problem With Jump Coefficients

In this paper, a type of arbitrary Lagrangian–Eulerian (ALE) finite element method in the monolithic frame is developed for a linearized fluid–structure interaction (FSI) problem — an unsteady Stokes/parabolic interface problem with jump coefficients and moving interface, where, the corresponding mixed finite element approximation in both semi- and fully discrete scheme are developed and analyzed based upon one type of ALE formulation and a novel H1- projection technique associated with a moving interface problem, and the stability and optimal convergence properties in the energy norm are obtained for both discretizations to approximate the solution of a transient Stokes/parabolic interface problem that is equipped with a low regularity. Numerical experiments further validate all theoretical results. The developed analytical approaches and numerical implementations can be similarly extended to a realistic FSI problem in the future

A Monolithic Arbitrary Lagrangian-Eulerian Finite Element Method for an Unsteady Stokes/Parabolic Interface Problem

In this thesis, a non-conservative arbitrary Lagrangian-Eulerian (ALE) method is developed and analyzed for a type of linearized Fluid-Structure Interaction (FSI) problem in a time dependent domain with a moving interface - an unsteady Stokes/parabolic interface problem with jump coefficients. The corresponding mixed finite element approximation is analyzed for both semi- and full discretization based upon the so-called non-conservative ALE scheme. The stability and optimal convergence properties in the energy norm are obtained for both schemes

A new three-dimensional mixed finite element for direct numerical simulation of compressible viscoelastic flows with moving free surfaces

The original publication is available at http://www.springer.comInternational audienceTA Mixed Finite Element (MFE) method for 3D non-steady flow of a viscoelastic compressible fluid is presented. It was used to compute polymer injection flows in a complex mold cavity, which involves moving free surfaces. The flow equations were derived from the Navier-Stokes incompressible equations, and we extended a mixed finite element method for incompressible viscous flow to account for compressibility (using the Tait model) and viscoelasticity (using a Pom-Pom like model). The flow solver uses tetrahedral elements and a mixed velocity/pressure/extra-stress/density formulation, where elastic terms are solved by decoupling our system and density variation is implicitly considered. A new DEVSS-like method is also introduced naturally from the MINI-element formulation. This method has the great advantage of a low memory requirement. At each time slab, once the velocity has been calculated, all evolution equations (free surface and material evolution) are solved by a space-time finite element method. This method is a generalization of the discontinuous Galerkin method, that shows a strong robustness with respect to both re-entrant corners and flow front singularities. Validation tests of the viscoelastic and free surface models implementation are shown, using literature benchmark examples. Results obtained in industrial 3D geometries underline the robustness and the efficiency of the proposed method

Numerical solution of Stefan problem with variable space grid method based on mixed finite element/finite difference approach

The purpose of this paper is to improve the accuracy and stability of the existing solutions to 1D Stefan problem with time-dependent Dirichlet boundary conditions. The accuracy improvement should come with respect to both temperature distribution and moving boundary location. The variable space grid method based on mixed finite element/finite difference approach is applied on 1D Stefan problem with time-dependent Dirichlet boundary conditions describing melting process. The authors obtain the position of the moving boundary between two phases using finite differences, whereas finite element method is used to determine temperature distribution. In each time step, the positions of finite element nodes are updated according to the moving boundary, whereas the authors map the nodal temperatures with respect to the new mesh using interpolation techniques. The authors found that computational results obtained by proposed approach exhibit good agreement with the exact solution. Moreover, the results for temperature distribution, moving boundary location and moving boundary speed are more accurate th an those obtained by variable space grid method based on pure finite differences. The authors’ approach clearly differs from the previous solutions in terms of methodology. While pure finite difference variable space grid method produces stable solution, the mixed finite element/finite difference variable space grid scheme is significantly more accurate, especially in case of high alpha. Slightly modified scheme has a potential to be applied to 2D and 3D Stefan problems.Accepted for publishin

Simulation of thin film flows with a moving mesh mixed finite element method

We present an efficient mixed finite element method to solve the fourth-order thin film flow equations using moving mesh refinement. The moving mesh strategy is based on harmonic mappings developed by Li et al. [J. Comput. Phys., 170 (2001), pp. 562-588, and 177 (2002), pp. 365-393]. To achieve a high quality mesh, we adopt an adaptive monitor function and smooth it based on a diffusive mechanism. A variety of numerical tests are performed to demonstrate the accuracy and efficiency of the method. The moving mesh refinement accurately resolves the overshoot and downshoot structures and reduces the computational cost in comparison to numerical simulations using a fixed mesh.Comment: 18 pages, 10 figure

Modeling of a Fluid-Structure Coupled System Using Port-Hamiltonian Formulation

The interactions between fluid and structural dynamics are an important subject of study in several engineering applications. In airplanes, for example, these coupled vibrations can lead to structural fatigue, noise and even instability. At ISAE, we have an experimental device that consists of a cantilevered plate with a fluid tank near the free tip. This device is being used for model validation and active control studies. This work uses the port-Hamiltonian systems formulation for modeling this experimental device. Structural dynamics and fluid dynamics are independently modeled as infinite-dimensional systems. The plate is approximated as a beam. Shallow water equations are used for representing the fluid in the moving tank. The global system is coupled and spatial discretization of the infinite-dimensional systems using mixed finite-element method allows to obtain a finite-dimensional system that is still Hamiltonian