Automatic grid construction for few-body quantum mechanical calculations

An algorithm for generating optimal nonuniform grids for solving the two-body Schr\"odinger equation is developed and implemented. The shape of the grid is optimized to accurately reproduce the low-energy part of the spectrum of the Schr\"odinger operator. Grids constructed this way are applicable to more complex few-body systems where the number of grid points is a critical limitation to numerical accuracy. The utility of the grid generation for improving few-body calculations is illustrated through an application to bound states of He trimers

Optimizing the geometrical accuracy of curvilinear meshes

This paper presents a method to generate valid high order meshes with optimized geometrical accuracy. The high order meshing procedure starts with a linear mesh, that is subsequently curved without taking care of the validity of the high order elements. An optimization procedure is then used to both untangle invalid elements and optimize the geometrical accuracy of the mesh. Standard measures of the distance between curves are considered to evaluate the geometrical accuracy in planar two-dimensional meshes, but they prove computationally too costly for optimization purposes. A fast estimate of the geometrical accuracy, based on Taylor expansions of the curves, is introduced. An unconstrained optimization procedure based on this estimate is shown to yield significant improvements in the geometrical accuracy of high order meshes, as measured by the standard Haudorff distance between the geometrical model and the mesh. Several examples illustrate the beneficial impact of this method on CFD solutions, with a particular role of the enhanced mesh boundary smoothness.Comment: Submitted to JC

AN ENHANCED WAVELET BASED METHOD FOR NUMERICAL SOLUTION OF HIGH ORDER BOUNDARY VALUE PROBLEMS

The Legendre wavelet collocation method (LWCM) is suggested in this study for solving high-order boundary value problems numerically. Eighth, tenth, and twelfth-order examples are used as test problems to ensure that the technique is efficient and accurate. In comparison to other approaches, the numerical results obtained using LWCM demonstrate that the method's accuracy is very good. The results indicate that the method requires less computational effort to achieve better results

Generalized partial differential equations for interactive design

This paper presents a method for interactive design by means of extending the PDE based approach for surface generation. The governing partial differential equation is generalized to arbitrary order allowing complex shapes to be designed as single patch PDE surfaces. Using this technique a designer has the flexibility of creating and manipulating the geometry of shape that satisfying an arbitrary set of boundary conditions. Both the boundary conditions which are defined as curves in 3-space and the spine of the corresponding PDE are utilized as interactive design tools for creating and manipulating geometry intuitively. In order to facilitate interactive design in real time, a compact analytic solution for the chosen arbitrary order PDE is formulated. This solution scheme even in the case of general boundary conditions satisfies exactly the boundary conditions where the resulting surface has an closed form representation allowing real time shape manipulation. In order to enable users to appreciate the powerful shape design and manipulation capability of the method, we present a set of practical examples

HIGH ORDER B-SPLINE COLLOCATION METHOD AND ITS APPLICATION FOR HEAT TRANSFER PROBLEMS

High order B-spline collocation for solving boundary value problem is presented in this paper. The approach employs high order B-spline basis functions with high approximation and continuity properties to handle problem domain with scattered or random distribution of knot points.  Using appropriate B-spline basis function construction, the new approach introduces no difficulties in imposing both Dirichlet and Neumann boundary conditions in the problem domain. Several numerical examples in arbitrary domains, both regular and irregular shaped domains, are considered in the present study. In addition, simulation results concerning with heat transfer applications are further presented and discussed

Extension of the Finite Integration Technique including dynamic mesh refinement and its application to self-consistent beam dynamics simulations

An extension of the framework of the Finite Integration Technique (FIT) including dynamic and adaptive mesh refinement is presented. After recalling the standard formulation of the FIT, the proposed mesh adaptation procedure is described. Besides the linear interpolation approach, a novel interpolation technique based on specialized spline functions for approximating the discrete electromagnetic field solution during mesh adaptation is introduced. The standard FIT on a fixed mesh and the new adaptive approach are applied to a simulation test case with known analytical solution. The numerical accuracy of the two methods are shown to be comparable. The dynamic mesh approach is, however, much more efficient. This is also demonstrated for the full scale modeling of the complete RF gun at the Photo Injector Test Facility DESY Zeuthen (PITZ) on a single computer. Results of a detailed design study addressing the effects of individual components of the gun onto the beam emittance using a fully self-consistent approach are presented.Comment: 33 pages, 14 figures, 4 table