Numerical approximation of internal discontinuity interface problems

This work focuses on the finite element discretization of boundary value problems whose solution features either a discontinuity or a discontinuous conormal derivative across an in- terface inside the computational domain. The interface is characterized via a level set function. The discontinuities are accounted for by using suitable extension operators whose numerical implementa- tion requires a very low computational effort. After carrying out the error analysis, numerical results to validate our approach are presented in one, two, and three dimensions.Peer ReviewedPostprint (published version

Finite Element Quadrature of Regularized Discontinuous and Singular Level Set Functions in 3D Problems

Regularized Heaviside and Dirac delta function are used in several fields of computational physics and mechanics. Hence the issue of the quadrature of integrals of discontinuous and singular functions arises. In order to avoid ad-hoc quadrature procedures, regularization of the discontinuous and the singular fields is often carried out. In particular, weight functions of the signed distance with respect to the discontinuity interface are exploited. Tornberg and Engquist (Journal of Scientific Computing, 2003,19: 527-552) proved that the use of compact support weight function is not suitable because it leads to errors that do not vanish for decreasing mesh size. They proposed the adoption of non-compact support weight functions. In the present contribution, the relationship between the Fourier transform of the weight functions and the accuracy of the regularization procedure is exploited. The proposed regularized approach was implemented in the eXtended Finite Element Method. As a three-dimensional example, we study a slender solid characterized by an inclined interface across which the displacement is discontinuous. The accuracy is evaluated for varying position of the discontinuity interfaces with respect to the underlying mesh. A procedure for the choice of the regularization parameters is propose

Numerical simulation of orbitally shaken viscous fluids with free surface

Orbitally shaken bioreactors are an emerging alternative to stirred-tank bioreactors for large-scale mam- malian cell culture, but their fluid dynamics is still not well defined. Among the theoretical and practical issues that remain to be resolved, the characterization of the liquid free surface during orbital shaking remains a major challenge because it is an essential aspect of gas transfer and mixing in these reactors. To simulate the fluid behavior and the free surface shape, we developed a numerical method based on the finite element framework. We found that the large density ratio between the liquid and the gas phases induced unphysical results for the free surface shape. We therefore devised a new pressure correction scheme to deal with large density ratios. The simulations operated with this new scheme gave values of wave amplitude similar to the ones mea- sured experimentally. These simulations were used to calculate the shear stress and to study the mixing principle in orbitally shaken bioreactorsPeer ReviewedPostprint (author’s final draft

Numerical Simulation of Orbitally Shaken Reactors

Mammalian cell cultures have become a major topic of research in the biopharmaceutical industry. This kind of cells requires specific conditions to grow. In this thesis, we study the hydrodynamics of orbitally shaken reactors (OSR), a recently introduced kind of bioreactors for mammalian cell cultures that represents a simple to operate and cheap alternative to commonly used reactors such as stirred tanks. OSRs can provide suitable conditions for small scale cell cultures, however a deeper understanding of the principles governing the OSRs is required to exploit their full potentiality and proceed with scaling up. This work aims at shedding light into the mechanisms of the OSRs through computational fluid dynamics. OSRs are only partially filled with liquid medium, the remaining space is occupied by air. When an OSR is agitated, the interface between the two phases moves and creates different shapes. This interface is at the heart of the simulation of OSRs: not only its location is part of the problem, but it can also carry singularities. In particular, the pressure has usually a low regularity in the vicinity of the interface and numerical methods might underperform if the singularity is not treated in an appropriate manner. This motivated the study of an elliptic problem in a medium with an internal interface carrying discontinuities. In this work, we devise a novel method called SESIC to solve this kind of problem. It uses the a priori knowledge to improve the numerical accuracy in the vicinity of the interface by removing the singularities. We prove that this method yields optimal orders of convergence in H1 and L2 norm. Numerical tests also show that optimal orders can be obtained in the L∞ norm in some cases. Regularized integration is also investigated with the perspective of further simplifying the scheme. It is found that, if the regularization bandwidth is suitably defined, good approximations can still be obtained, even if the convergence rate is decreased. We apply then the methodology of the SESIC method to the approximation of the two-phase Navier-Stokes equations, which amounts to correct the pressure. If adapted integration is used with it, the density and viscosity can be kept discontinuous across the interface without creating spurious velocity, as shown by numerical experiments. The sharp treatment of the discontinuities improves the accuracy of the simulations by retaining the physical meaning of the phenomena independently of the mesh size. We also pay attention to the boundary conditions used, which must be suitably chosen to allow the interface motion but still reproduce the wall friction. We show that imposing the zero normal component of the velocity yields the best results for the no-penetration condition and that it must be employed with a correction term to avoid spurious velocities. Robin-type conditions are used for the tangential components to recover the no-slip condition far from the contact line. Specific tests are performed to assess the quality of the different components of the method. We also compare it with a regularized density/viscosity method and show that the sharp treatment of these physical quantities improves the quality of the simulation. The scalability properties of the method are also investigated and the bottlenecks pointed out. Our method is then validated in various ways with experimental data. First of all, glycerine filled OSRs are simulated and we show that our method reproduces accurately the amplitude of the generated wave. The sensibility of the results with respect to the Robin condition is shown to be weak. We investigate then water filled OSRs. The different wave patterns, either breaking or non-breaking, single or multiple, observed experimentally are reproduced for various configurations. In particular, triple waves are obtained as well. We use laser Doppler velocimetry measures of the velocity field to further validate our simulations. The hydrodynamic stress and the mixing pattern of the different regimes are evaluated and put into relationship with the wave shape. Finally, we investigate the modelling of the cell culture by devising a system non-linear ODEs which represents the evolution in time of the main nutriments and wastes and the growth of the cell population. The behaviour of the cell culture is well reproduced, but some phenomena remain unexplained. In particular, our model contains a toxic waste whose actual identity is discussed. Two alternative scenarios are proposed to improve the model

Numerical approximation of internal discontinuity interface problems

This work focuses on the finite element discretization of boundary value problems whose solution features either a discontinuity or a discontinuous conormal derivative across an in- terface inside the computational domain. The interface is characterized via a level set function. The discontinuities are accounted for by using suitable extension operators whose numerical implementa- tion requires a very low computational effort. After carrying out the error analysis, numerical results to validate our approach are presented in one, two, and three dimensions.Peer Reviewe