Boundary element methods discretization technique

The Mie theory [1] and the T-matrix method [4] are very efficient for (multilayered) spheres and axisymmetric particles (with moderate aspect ratios), respectively. Several methods, applicable to particles of arbitrary shapes, have been used in plasmonic simulations the boundary element method (BEM) [5, 6], the DDA [7-9], the finite-difference time-domain method (FDTD) [10, 11], the finite element method (FEM] [12,13], the finite integration technique (FIT) [14] and the null-field method with discrete sources (NFM-DS) [15,16]. There is also quasi-static approximation for spheroids [12], but it is not discussed here. 

When non-linearities are included in the analysis, we must also solve the domain integral in the integral formulations. Several methods have been developed to approximate this integral. As a matter of fact, at the international conferences on boundary elements, organized every year since 1978 [43], numerous papers on different and novel techniques to approximate the domain integral have been presented in order to make the BEM applicable to complex non-linear and time dependent problems. Many of these papers were pointing out the difficulties of extending the BEM to such applications. The main drawback in most of the techniques was the need to discretize the domain into a series of internal cells to deal with the terms not taken to the boundary by application of the fundamental solution, such as non-linear terms. 

Once we have stated or derived the mathematical equations which define the physics of the system, we must figure out how to solve these equations for the particular domain we are interested in. Most numerical methods for solving boundary value problems require that the continuous domain be broken up into discrete elements, the so-called mesh or grid, which one can use to approximate the governing equation (s) using the particular numerical technique (finite element, boundary element, finite difference, or multigrid) best suited to the problem. 

The finite element method is a powerful technique for finding approximate solution of a partial differential equation where the domain boundaries of a given problem are complex [16]. It has now become one of the fundamental numerical approaches for solving problems arising in many applications, including biomedical simulation. In the finite element method, a complex domain is discretized into a number of elements, such as that a set of basis functions can be defined on the elements to approximate the solution [8]. 

Once the seismic source and Earth crustal model have been adequately described, nearfault ground motion simulations in the low-frequency range (e.g., below 1 Hz) can be performed using deterministic modeling techniques [e.g., discrete wavenumber method (DWN), finite difference method (FDM), finite element method (FEM), boundary element... 

The flow of many red blood cells in wider capillaries has also been investigated by several simulation techniques. Discrete fluid-particle simulations - an extension of DPD - in combination with bulk-elastic discocyte cells (in contrast to the membrane elasticity of real red blood cells) have been employed to investigate the dynamical clustering of red blood cells in capillary vessels [223,224], An immersed finite-element model - a combination of the immersed boundary method for the solvent hydrodynamics [225] and a finite-element method to describe the membrane elasticity - has been developed to study red blood cell aggregation [226]. Finally, it has been demonstrated that the LB method for the solvent in combination with a triangulated mesh model with curvature and shear elasticity for the membrane can be used efficiently to simulate RBC suspensions in wider capillaries [189]. 

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

To reduce this effort, the software Polyflow (Fluent, Lebanon, USA) contains a special module to avoid the remeshing of the flow channel for every single timestep. This is called the Mesh Superposition Technique , where the inner barrel and the screw are meshed separatly. The discrete meshes are overlayed to create one system where the surfaces of the screw define the channel boundary. A major issue with this method is that the flow channel volume varies as the intersection of the surface elements leads to unequal sums over all elements. This is compensated by a compression factor on which the simulation results react very sensitively. 

Meshless methods belong to a class of techniques for solving boundary/initial value partial differential equations in which both geometry representation and numerical discretization are principally performed based on nodes or particles. In meshless methods, there is no inherent reliance on a particular mesh topology, meaning that no element connectivity is required. In practice, however, in many meshless methods, recourse must be taken to some kind of background meshes at least in one stage of the implementation. 

---