The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

高等学校計算数学学報 = Numerical mathematics

An efficient ghost-cell immersed boundary method (GCIBM) for simulating turbulent flows in complex geometries is presented. A boundary condition is enforced through a ghost cell method. The reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second-order accuracy of the base solver. Both Dirichlet and Neumann boundary conditions can be treated. The current ghost cell treatment is both suitable for staggered and non-staggered Cartesian grids. The accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface. Numerical results are compared with experimental data and boundary-fitted grid results. The method is further extended to an existing ocean model (MITGCM) to simulate geophysical flow over a three-dimensional bump. The method is easily implemented as evidenced by our use of several existing codes.

A Ghost-Cell Immersed Boundary Method for Flow in Complex Geometry

The interactions between incompressible fluid flows and immersed struc- tures are nonlinear multi-physics phenomena that have applications to a wide range of scientific and engineering disciplines In this article, we review representative numeri- cal methods based onconforming and non-conforming meshes that arecurrentlyavail- able for computing fluid-structure interaction problems, with an emphasis on some of the recent developments in the field A goal is to categorize the selected methods and assess their accuracy and efficiency We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study in fluid-structure interactions

/pdf/numerical-methods-for-fluid-structure-interaction-a-review-2raeajc0ja.pdf

Numerical Methods for Fluid-Structure Interaction — A Review

A simple yet effective modification to the standard finite element method is presented in this paper. The basic idea is an extension of a partial differential equation beyond the physical domain of computation up to the boundaries of an embedding domain, which can easier be meshed. If this extension is smooth, the extended solution can be well approximated by high order polynomials. This way, the finite element mesh can be replaced by structured or unstructured cells embedding the domain where classical h- or p-Ansatz functions are defined. An adequate scheme for numerical integration has to be used to differentiate between inside and outside the physical domain, very similar to strategies used in the level set method. In contrast to earlier works, e.g., the extended or the generalized finite element method, no special interpolation function is introduced for enrichment purposes. Nevertheless, when using p-extension, the method shows exponential rate of convergence for smooth problems and good accuracy even in the presence of singularities. The formulation in this paper is applied to linear elasticity problems and examined for 2D cases, although the concepts are generally valid.

Finite cell method

https://hal.archives-ouvertes.fr/hal-00274445/document

A Fictitious domain approach with spread interface for elliptic problems with general boundary conditions

https://hal.archives-ouvertes.fr/hal-00274475/document

A general fictitious domain method with immersed jumps and multilevel nested structured meshes

Fixed point iterations are still the most common approach to dealing with a variety of numerical problems such as coupled problems (multi-physics, domain decomposition, ?) or nonlinear problems (electronic structure, heat transfer, nonlinear mechanics, ?). Methods to accelerate fixed point iteration convergence or more generally sequence convergence have been extensively studied since the 1960's. For scalar sequences, the most popular and efficient acceleration method remains the Δ 2 of Aitken. Various vector acceleration algorithms are available in the literature, which often aim at being multi-dimensional generalizations of the Δ 2 method.In this paper, we propose and analyse a generic residual-based formulation for accelerating vector sequences. The question of the dynamic use of this residual-based transformation during the fixed point iterations for obtaining a new accelerated fixed point method is then raised. We show that two main classes of such iterative algorithms can be derived and that this approach is generic in that various existing acceleration algorithms for vector sequences are thereby recovered.In order to illustrate the interest of such algorithms, we apply them in the field of nonlinear mechanics on a simplified "point-wise" solver used to perform mechanical behaviour unit testings. The proposed test cases clearly demonstrate that accelerated fixed point iterations based on the elastic operator (quasi-Newton method) are very useful when the mechanical behaviour does not provide the so-called consistent tangent operator. Moreover, such accelerated algorithms also prove to be competitive with respect to the standard Newton-Raphson algorithm when available.

https://hal-cea.archives-ouvertes.fr/cea-01403292/document

Iterative residual-based vector methods to accelerate fixed point iterations

SUMMARY The aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain’s boundary. This problematic of the last century (e.g. the finite difference methodology) returns to topical studies with the huge development of domain embedding, fictitious domain or Cartesiangrid methods. These methods use regular structured meshes (most often Cartesian) for non-aligned domains. Although non-boundary fitted approaches become more and more applied, very few studies are devoted to theoretical error estimates. In this paper, the convergence of a Q1-nonconforming finite element method is analyzed for secondorder elliptic problems with Dirichlet, Robin or Neumann boundary conditions. The finite element method uses standard Q1 rectangular finite elements. As the finite element approximate space is not contained in the original solution space, this method is referred to as nonconforming. A stairstep boundary defined from the Cartesian mesh approximates the original domain’s boundary. The convergence analysis of the finite element method for such a kind of non-boundary fitted stairstepped approximation is no treated in the literature. The study of Dirichlet problems is based on similar techniques as those classically used with boundary-fitted linear triangular finite elements. The estimates obtained for Robin problems are novel and use some more technical arguments. The rate of convergence is proved to be in O(h 1/2 ) for the H 1 norm for all general boundary conditions, and classical duality arguments allow to obtain an O(h) error estimate in the L 2 norm for Dirichlet problems. Numerical results obtained with fictitious domain techniques, that impose original boundary conditions on a non-boundary fitted approximate immersed interface, are presented. These results confirm the theoretical rates of convergence. Copyright c 2007 John Wiley & Sons, Ltd.

Isabelle Ramière

Papers

A Fictitious domain approach with spread interface for elliptic problems with general boundary conditions

A general fictitious domain method with immersed jumps and multilevel nested structured meshes

Iterative residual-based vector methods to accelerate fixed point iterations

Convergence analysis of the Q1-finite element method for elliptic problems with non-boundary-fitted meshes

GERMINAL, a fuel performance code of the PLEIADES platform to simulate the in-pile behaviour of mixed oxide fuel pins for sodium-cooled fast reactors