Global Basis Functions Research Articles

Meshless physical simulation of semiconductor devices using a wavelet-based nodes generator

This paper describes a meshless method with wavelet-based nodes for the two-dimensional time-dependent simulation of semiconductor devices. In this method the solution is approximated using global radial basis functions (RBF) and distributed wavelet-generated points. This allows the computation of problems with complex-shaped boundaries and forming fine and coarse points abundance in locations where variable solutions change rapidly and slowly, respectively. The method is suitable for the semiconductor part of very time consuming global modeling of microwave/millimeter wave circuits due to a large reduction of number of nodes with an acceptable results.

Eigenvalue Analysis and Longtime Stability of Resonant Structures for the Meshless Radial Point Interpolation Method in Time Domain

A meshless collocation method based on radial basis function (RBF) interpolation is presented for the numerical solution of Maxwell's equations. RBFs have attractive properties such as theoretical exponential convergence for increasingly dense node distributions. Although the primary interest resides in the time domain, an eigenvalue solver is used in this paper to investigate convergence properties of the RBF interpolation method. The eigenvalue distribution is calculated and its implications for longtime stability in time-domain simulations are established. It is found that eigenvalues with small, but nonzero, real parts are related to the instabilities observed in time-domain simulations after a large number of time steps. Investigations show that by using global basis functions, this problem can be avoided. More generally, the connection between the high matrix condition number, accuracy, and the magnitude of nonzero real parts is established.

Electromagnetic Modeling of Through-Silicon Via (TSV) Interconnections Using Cylindrical Modal Basis Functions

This paper proposes an efficient method to model through-silicon via (TSV) interconnections, an essential building block for the realization of silicon-based 3-D systems. The proposed method results in equivalent network parameters that include the combined effect of conductor, insulator, and silicon substrate. Although the modeling method is based on solving Maxwell's equation in integral form, the method uses a small number of global modal basis functions and can be much faster than discretization-based integral-equation methods. Through comparison with 3-D full-wave simulations, this paper validates the accuracy and the efficiency of the proposed modeling method.

Dimension-splitting data points redistribution for meshless approximation

To better approximate nearly singular functions with meshless methods, we propose a data points redistribution method extended from the well-known one-dimensional equidistribution principle. With properly distributed data points, nearly singular functions can be well approximated by linear combinations of global radial basis functions. The proposed method is coupled with an adaptive trial subspace selection algorithm in order to reduce computational cost. In our numerical examples, clear exponential convergence (with respect to the numbers of data points) can be observed.

A spectral boundary integral method for flowing blood cells

A spectral boundary integral method for simulating large numbers of blood cells flowing in complex geometries is developed and demonstrated. The blood cells are modeled as finite-deformation elastic membranes containing a higher viscosity fluid than the surrounding plasma, but the solver itself is independent of the particular constitutive model employed for the cell membranes. The surface integrals developed for solving the viscous flow, and thereby the motion of the massless membrane, are evaluated using an O ( N log N ) particle-mesh Ewald (PME) approach. The cell shapes, which can become highly distorted under physiologic conditions, are discretized with spherical harmonics. The resolution of these global basis functions is, of course, excellent, but more importantly they facilitate an approximate de-aliasing procedure that stabilizes the simulations without adding any numerical dissipation or further restricting the permissible numerical time step. Complex geometry no-slip boundaries are included using a constraint method that is coupled into an implicit system that is solved as part of the time advancement routine. The implementation is verified against solutions for axisymmetric flows reported in the literature, and its accuracy is demonstrated by comparison against exact solutions for relaxing surface deformations. It is also used to simulate flow of blood cells at 30% volume fraction in tubes between 4.9 and 16.9 μm in diameter. For these, it is shown to reproduce the well-known non-monotonic dependence of the effective viscosity on the tube diameter.

Galerkin orthogonal polynomials

Abstract The Galerkin method offers a powerful tool in the solution of differential equations and function approximation on the real interval [−1, 1]. By expanding the unknown function in appropriately chosen global basis functions, each of which explicitly satisfies the given boundary conditions, in general this scheme converges exponentially fast and almost always supplies the most terse representation of a smooth solution. To date, typical schemes have been defined in terms of a linear combination of two Jacobi polynomials. However, the resulting functions do not inherit the expedient properties of the Jacobi polynomials themselves and the basis set will not only be non-orthogonal but may, in fact, be poorly conditioned. Using a Gram-Schmidt procedure, it is possible to construct, in an incremental fashion, polynomial basis sets that not only satisfy any linear homogeneous boundary conditions but are also orthogonal with respect to the general weighting function ( 1 - x ) α ( 1 + x ) β . However, as it stands, this method is not only cumbersome but does not provide the structure for general index n of the functions and obscures their dependence on the parameters ( α , β ) . In this paper, it is shown that each of these Galerkin basis functions, as calculated by the Gram-Schmidt procedure, may be written as a linear combination of a small number of Jacobi polynomials with coefficients that can be determined. Moreover, this terse analytic representation reveals that, for large index, the basis functions behave asymptotically like the single Jacobi polynomial P n ( α , β ) ( x ) . This new result shows that such Galerkin bases not only retain exponential convergence but expedient function-fitting properties too, in much the same way as the Jacobi polynomials themselves. This powerful methodology of constructing Galerkin basis sets is illustrated by many examples, and it is shown how the results extend to polar geometries. In exploring more generalised definitions of orthogonality involving derivatives, we discuss how a large class of differential operators may be discretised by Galerkin schemes and represented in a sparse fashion by the inverse of band-limited matrices.

A quasi optimal Petrov–Galerkin method for Helmholtz problem

AbstractA Petrov–Galerkin finite element formulation is introduced for Helmholtz problem in two dimensions using polynomial weighting functions. At each node of the mesh, a global basis function for the weighting space is obtained adding to the bilinear C0 Lagrangian weighting function linear combinations of polynomial bubbles defined on a macroelement containing this node. Quasi optimal weighting functions, with the same support of the corresponding global test functions, are obtained after computing the coefficients of these linear combinations attending to optimality criteria. This is done numerically through a simple preprocessing technique naturally applied to non‐uniform and unstructured meshes. In particular, for uniform mesh a quasi optimal interior stencil of the same order of the quasi‐stabilized finite element method stencil derived by Babuška et al. (Comput. Methods Appl. Mech. Engrg 1995; 128:325–359) is obtained. Numerical results are presented illustrating the great stability and accuracy of this formulation with uniform and non‐uniform meshes. Copyright © 2009 John Wiley & Sons, Ltd.

Fast radial basis function interpolation with Gaussians by localization and iteration

Radial basis function (RBF) interpolation is a technique for representing a function starting with data on scattered points. In the vortex particle method for solving the Navier–Stokes equations, the representation of the field of interest (the fluid vorticity) as a sum of Gaussians is like an RBF approximation. In this application, there are two instances when one may need to solve an RBF interpolation problem: upon initialization of the vortex particles, and after spatial adaptation to replace a set of disordered particles by another set with a more regular distribution. Solving large RBF interpolation problems is notoriously difficult with basis functions of global support, due to the need to solve a linear system with a fully populated and badly conditioned matrix. In the vortex particle method, in particular, one uses Gaussians of very small spread, which lead us to formulate a method of solution consisting of localization of the global problem, and improvement of solutions over iterations. It can be thought of as an opposite approach from the use of compact support basis functions to ease the solution procedure. Compact support bases result in sparse matrices, but at the cost of reduced approximation qualities. Instead, we keep the global basis functions, but localize their effect during the solution procedure, then add back the global effect of the bases via the iterations. Numerical experiments show that convergence is always obtained when the localized domains overlap via a buffer layer, and the particles overlap moderately. Algorithmic efficiency is excellent, achieving convergence to almost machine precision very rapidly with well-chosen parameters. We also discuss the complexity of the method and show that it can scale as O(N) for N particles.

A least-squares approximation of partial differential equations with high-dimensional random inputs

Uncertainty quantification schemes based on stochastic Galerkin projections, with global or local basis functions, and also stochastic collocation methods in their conventional form, suffer from the so called curse of dimensionality: the associated computational cost grows exponentially as a function of the number of random variables defining the underlying probability space of the problem. In this paper, to overcome the curse of dimensionality, a low-rank separated approximation of the solution of a stochastic partial differential (SPDE) with high-dimensional random input data is obtained using an alternating least-squares (ALS) scheme. It will be shown that, in theory, the computational cost of the proposed algorithm grows linearly with respect to the dimension of the underlying probability space of the system. For the case of an elliptic SPDE, an a priori error analysis of the algorithm is derived. Finally, different aspects of the proposed methodology are explored through its application to some numerical experiments.

MULTIPLE CAVITY MODELING OF A FEED NETWORK FOR TWO DIMENSIONAL PHASED ARRAY APPLICATION

In this paper a method of moment based analysis of an H-plane 1:3power divider has been presented using Multi Cavity Modeling Technique (MCMT) in transmitting mode. Finally attempt has been made to improve the frequency response characteristic of the above mentioned waveguide circuit using a sorting post to diversify the power equally in all the ports. Codes have been written for analyzing the frequency response characteristic of the structure, mentioned above. Numerical data have been compared with the data obtained with laboratory measurement, and CST Microwave Studio simulation. In the present analysis global basis function has been used. The existence of cross polarization components of the field inside the waveguide structures if exists, have also been considered to obtain an accurate result. The proposed power divider has good agreement with the theoretical; CST microwave studio simulated data and measured data. The power divider can be used in the input at 9.8 GHz frequency band, over 800 MHz.

SU‐FF‐I‐70: An Efficient Deformable Volume Rendering On Modern PC Graphics Hardware

Purpose: Direct volume rendering (DVR) is a common medical image visualization technique that provides better anatomy or object perception in three dimensional space than typical three orthogonal sectional images. In this study we present a fast and efficient deformable DVR method, implemented on modern consumer grade PC graphics hardware, which will be useful for intensity‐based deformable 3D/2D registrations as well as for deforming volume visualization during 3D/3D registrations. Method and Materials: The method consists of three steps: (1) a cubical grid is defined on the input volume; (2) the grid is deformed using global or local radial basis functions such as thin plate splines; and then (3) the volume is texture‐mapped to the deformed grid and projected to a predefined 2D pixel buffer. During projection, pixel intensities are scaled and accumulated using either attenuate or over blending operation. To minimize the intensity depth loss during the accumulation process, the pixel intensities are accumulated in a hidden pixel buffer, known as pbuffer in OpenGL. In our implementation we used 16 bit floating pbuffer instead of the standard 32 bit floating format because most graphics hardware did not support direct accumulation onto 32 bit floating buffers at the time of experiment. The accumulated intensity values are scaled to the valid display range [0,1]. Results: We were able to render a 256×256×256 anthropomorphic chest phantom CT image onto 512×512 render buffer with 20×20×20 grid at the speed of about 3 frames per second. The computer was equipped with a 1.8 GHz CPU, 512MB memory, 4× AGP, and an NVidia GeForce 6800 graphics adaptor. Conclusion: Deformable DVR on modern PC graphics hardware is efficient in terms of speed and quality. This method may be useful for deformable 3D/2D image registrations and visualization for 3D/3D image registrations.

The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions

In this paper a spectral method and a numerical continuation algorithm for solving eigenvalue problems for the rectangular von Kármán plate with different boundary conditions (simply supported, partially or totally clamped) and physical parameters are introduced. The solution of these problems has a postbuckling behaviour. The spectral method is based on a variational principle (Galerkin’s approach) with a choice of global basis functions which are combinations of trigonometric functions. Convergence results of this method are proved and the rate of convergence is estimated. The discretized nonlinear model is treated by Newton’s iterative scheme and numerical continuation. Branches of eigenfunctions found by the algorithm are traced. Numerical results of solving the problems for polygonal and ferroconcrete plates are presented.

An Improved Multiquadric Collocation Method for 3-D Electromagnetic Problems

The multiquadric radial basis function method (MQ RBF or, simply, MQ) developed recently is a truly meshless collocation method with global basis functions. It was introduced for solving many 1- and 2-D partial differential equations (PDEs), including linear and nonlinear problems. However, few works are found for electromagnetic PDEs, especially for 3-D problems. This paper presents an improved MQ collocation method for 3-D electromagnetic problems. Numerical results show a considerable improvement in accuracy over the traditional MQ collocation method, although both methods are direct collocation method with exponential convergence

Eddy currents in a gradient coil, modeled as circular loops of strips

As a model for the z-coil of an MRI-scanner a set of circular loops of strips, or rings, placed on one cylinder is chosen. The current in this set of thin conducting rings is driven by an external source current. The source, and all excited fields, are time-harmonic. The frequency is low enough to allow for an electro-quasi-static approach. The rings have a thin rectangular cross-section with a thickness so small that the current can be assumed uniformly distributed in the thickness direction. Due to induction, eddy currents occur resulting in an edge-effect. Higher frequencies cause stronger edge-effects. As a consequence, the resistance of the system increases and its self-inductance decreases. The Maxwell equations imply an integral equation for the current distribution in the rings. The Galerkin method with Legendre polynomials as global basis functions is applied. This method shows fast convergence, so only a very restricted number of basis functions is needed. The general method is worked out for N (N ≥ 1) rings, and explicit results are presented for N = 1, N = 2 and N = 24. The derived integral equation and the numerical results of the simulations show that sets of circular rings and plane strips describe the same electromagnetic behavior, thus demonstrating that inductance effects are local.

Free Vibration Analysis of Composite and Sandwich Plates by a Trigonometric Layerwise Deformation Theory and Radial Basis Functions

In this study trigonometric layerwise deformation theory is used for the analysis of free vibration of symmetric composite plates. A meshless discretization method based on global multiquadric radial basis functions is used. The equations of motion and the boundary conditions are derived and interpolated by radial basis functions. This method is applied to the free vibration analysis of composite and sandwich plates. The results are then compared with analytical and numerical solutions. The results show that the use of trigonometric layerwise deformation theory discretized with multiquadrics provides very good solutions for the free vibration of composite and sandwich plates.

A radial basis collocation method for Hamilton–Jacobi–Bellman equations

In this paper we propose a semi-meshless discretization method for the approximation of viscosity solutions to a first order Hamilton–Jacobi–Bellman (HJB) equation governing a class of nonlinear optimal feedback control problems. In this method, the spatial discretization is based on a collocation scheme using the global radial basis functions (RBFs) and the time variable is discretized by a standard two-level time-stepping scheme with a splitting parameter θ . A stability analysis is performed, showing that even for the explicit scheme that θ = 0 , the method is stable in time. Since the time discretization is consistent, the method is also convergent in time. Numerical results, performed to verify the usefulness of the method, demonstrate that the method gives accurate approximations to both of the control and state variables.

COMPARATIVE STUDY OF THE MULTIQUADRIC AND THIN-PLATE SPLINE RADIAL BASIS FUNCTIONS FOR THE TRANSIENT-CONVECTIVE DIFFUSION PROBLEMS

The numerical solutions of the unsteady transient-convective diffusion problems are investigated by using multiquadric (MQ) and thin-plate spline (TPS) radial basis functions (RBFs) based on mesh-free collocation methods with global basis functions. The results of radial basis functions are compared with the mesh-dependent boundary element and finite difference methods as well as the analytical solution for high Péclet numbers. It is reported that for low Péclet numbers, MQ-RBF provides excellent agreement, while for high Péclet numbers, TPS-RBF is better than MQ-RBF.

Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.

Nonlinear Modeling of Mechanical Gas Face Seal Systems Using Proper Orthogonal Decomposition

An approach based on proper orthogonal decomposition and Galerkin projection is presented for developing low-order nonlinear models of the gas film pressure within mechanical gas face seals. A technique is developed for determining an optimal set of global basis functions for the pressure field using data measured experimentally or obtained numerically from simulations of the seal motion. The reduced-order gas film models are shown to be computationally efficient compared to full-order models developed using the conventional semidiscretization methods. An example of a coned mechanical gas face seal in a flexibly mounted stator configuration is presented. Axial and tilt modes of stator motion are modeled, and simulation studies are conducted using different initial conditions and force inputs. The reduced-order models are shown to be applicable to seals operating within a wide range of compressibility numbers, and results are provided that demonstrate the global reduced-order model is capable of predicting the nonlinear gas film forces even with large deviations from the equilibrium clearance.

Static and free vibration analysis of composite shells by radial basis functions

The higher-order shear deformation theory of laminated orthotropic elastic shells of Reddy accounts for parabolic distribution of the transverse shear strains through the thickness of the shell. The Reddy shell theory allows the fulfillment of homogeneous conditions (zero values) at the top and bottom surfaces of the shell. This paper deals with a meshless solution of the Reddy higher order shell theory in static and free vibration analysis. The meshless technique is based on the asymmetric global multiquadric radial basis function method proposed by Kansa. This paper demonstrates that this truly meshless method is very successful in the static and free vibration analysis of laminated composite shells.