Double-layer Formulation Research Articles

Immersed Boundary Double Layer method: An introduction of methodology on the Helmholtz equation

The Immersed Boundary (IB) method of Peskin (1977) [1] is useful for problems that involve fluid-structure interactions or complex geometries. By making use of a regular Cartesian grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. Additionally, the IB method has been adapted to problems with prescribed motion and other PDEs with given boundary data. IB methods for these problems traditionally involve penalty forces which only approximately satisfy boundary conditions, or they are formulated as constraint problems. In the latter approach, one must find the unknown forces by solving an equation that corresponds to a poorly conditioned first-kind integral equation. This operation can therefore require a large number of iterations of a Krylov method, and since a time-dependent problem requires this solve at each step in time, this method can be prohibitively inefficient without preconditioning. This work introduces a new, well-conditioned IB formulation for boundary value problems, called the Immersed Boundary Double Layer (IBDL) method. In order to lay the groundwork for similar formulations of Stokes and Navier-Stokes equations, this paper focuses on the Poisson and Helmholtz equations to introduce the methodology and to demonstrate its efficiency over the original constraint method. In this double layer formulation, the equation for the unknown boundary distribution corresponds to a well-conditioned second-kind integral equation that can be solved efficiently with a small number of iterations of a Krylov method without preconditioning. Furthermore, the iteration count is independent of both the mesh size and spacing of the immersed boundary points. The method converges away from the boundary, and when combined with a local interpolation, it converges in the entire PDE domain. Additionally, while the original constraint method applies only to Dirichlet problems, the IBDL formulation can also be used for Neumann boundary conditions.

“Application of boundary integral solution to thermal diffusion problem by edge based basis functions”

In this research paper, heat diffusion problem is solved using the new numerical simulation method called boundary integral equation formulation and boundary element solution method. Three dimensional irregular thermal bodies are addressed by introducing the new edge based basis and weighting functions. The three numerical formulations defined on the edges are Single Layer Formulation (EBSLF), Double Layer Formulation (EBDLF) and Combined Layer Formulation (EBCLF) are presented for higher accuracy of the results for heat dissipation. The triangular patch modeling method is used to divide the arbitrary irregular geometry of the body into triangular patches so that above edge based function are defined on those edge of the patches for numerical solution. Numerically, the solution method is developed by defining the basis functions on the Edge of the triangles, instead of on the nodes as followed in the regular BEM solutions. The temperature diffusion results from the hot surface body are plotted for analysis. Also the convergence study is done to get the result convergence towards the exact solution for the different new numerical formulations. The errors are treated in two different ways. The actual error and the absolute errors are used to determine the best model among all the combinations of numerical formulations and different mesh sizes.

A paradigm shift in enteric coating: Achieving rapid release in the proximal small intestine of man

The in vivo performance of a novel enteric double-coating technology designed to accelerate release in the proximal small intestine of humans was investigated. Tablet cores were coated with a double layer formulation consisting of an inner layer of EUDRAGIT® L 30 D-55 neutralised to pH 6.0 in the presence of 10% citric acid, and an outer layer of standard EUDRAGIT® L 30 D-55. A conventional single coating of EUDRAGIT® L 30 D-55 was also applied to tablets for comparison purposes, with the identical coating formulation and thickness (5 mg/cm 2) as the outer layer of the double coating. Eight fasted volunteers received the double-coated and single-coated tablets in a two-way crossover study. The formulations were radiolabelled and followed by gamma scintigraphy; the disintegration times and positions were recorded. After leaving the stomach, tablets coated with the single-coating formulation showed a significant time delay before disintegration occurred in the mid to distal small intestine, with a mean disintegration time of 66 ± 22 min post gastric emptying. The double-coated tablets disintegrated earlier at 28 ± 6 min post gastric emptying with consistent disintegration in the proximal small intestine. The accelerated in vivo disintegration of the double-coating system can overcome the limitations of conventional enteric coatings.

A Faster Method of Moments Solution to Double Layer Formulation of Acoustic Scattering

In this work, the acoustic scattering problem based oil double layer formulation is solved with a novel numerical technique using method of moment's solution. A new set of basis functions, namely, Edge based Adaptive Basis Functions (EABF) are defined in the method of moment's solution procedure. The geometry of the body is divided into triangular patches and basis functions are defined oil the edges. Since the double layer formulation involves the evaluation of the hyper-singular integral, the edge based adaptive basis functions are used to make the solution faster. The matrix equations are derived for the double layer formulation. The edge based adaptive basis functions used in this work generates a diagonal moment matrix and hence do not need any matrix inversion to be carried out. The scattering cross section of the canonical shapes is used to validate the numerical solution developed and the plots are presented.

Node based Method of Moments Solution to Combined Layer Formulation of Acoustic Scattering

In this work, a novel numerical technique, based on method of moments Solution, Is presented to solve the Combined layer formulation (CLF) to insure unique Solution to the exterior acoustic scattering problem Lit all frequencies. A new set of basis functions, narnely, Node based basis functions are used to represent the Source distribution Oil the Surface of rigid body and the same functions are used as testing, functions as well. Combined layer formulation (CLF) is defined by linearly combining the Single layer formulation (SLF) and Double layer formulation (DLF) with complex coupling parameter The matrix equations for the SLF and DLF are derived and same equations are oxtencled to the CLE The surface of the body is modeled by triangular patch modeling. Results of numerical technique presented in this paper, using node based basis functions, are compared with the exact Solutions wherever available. Also, condition numbers of the impedance matrices of SLF, DLF and CLF are plotted, for some objects of canonical shapes, to show that only CLF is free from resonance problem.

Elimination of internal resonance problem associated with acoustic scattering by three-dimensional rigid body

In this work, a simple and stable numerical method is presented, utilizing the method of moments (MoM), to eliminate the internal resonance problem associated with acoustic scattering by three-dimensional rigid body subjected to a plane wave incidence. The numerical method is based on the potential theory and combines the single layer formulation (SLF) and the double layer formulation (DLF). The scattering body is approximated by planar triangular patches. For the MoM solution of SLF and DLF, the basis functions have been defined with respect to the edges to approximate the unknown source distribution. These basis functions along with an efficient testing procedure generate accurate results at all frequencies, including the characteristic frequencies. Finally, the new solution method is validated with several representative examples.

Acoustic scattering from rigid bodies of arbitrary shape—Double layer formulation

In this work, a numerical method is presented, based on the well-known method of moments (MoM), to calculate the acoustic fields scattered by a three-dimensional, arbitrarily shaped, acoustically rigid body subjected to a plane wave incidence. The mathematical formulation is based on the potential theory and, in this work, the numerical method is applicable to the so-called double layer formulation (DLF). The scattering body is approximated by planar triangular patches. For the MoM solution using triangular patch modeling, edge-based basis functions are utilized to approximate the source distribution efficiently. These basis functions along with an appropriate testing procedure generates a simple numerical algorithm that is versatile enough to be applicable to closed, as well as open bodies. Finally, the present solution method is validated with several representative examples.