Parallel pseudospectral domain decomposition techniques

The influence of interface boundary conditions on the ability to parallelize pseudospectral multidomain algorithms is investigated. Using the properties of spectral expansions, a novel parallel two-domain procedure is generalized to an arbitrary number of domains each of which can be solved on a separate processor. This interface boundary condition considerably simplifies influence matrix techniques.     

Domain decomposition and the numerical solution of partial differential equations defined on irregular domains using segmented forms of the tau lines method

The Tau Lines Method, a numerical technique based on the combination of the Tau Method and the Method of Lines is used, in connection with the domain decomposition technique, to solve problems in partial differential equations defined on irregular domains. Two nontrivial problems have been considered. The first is a curved crack defined on a square domain and the second is defined on a kite-shaped domain. The domain of interest is subdivided into appropriate Semidiscretized elements so to efficiently deal with any appearance of boundary and/or interior singularities. Numerical application is carried out on the Poisson's equation. The approximate solutions are sought along segmented lines as finite expansions in terms of a given orthogonal polynomial basis. Two types of orthogonal expansions have been used alongside second and fourth order accurate Finite Difference Approximations. In both cases good and rapid convergence has been achieved.     

Hybrid spectral-element-low-order methods for incompressible flows

In this article we present a new formulation for coupling spectral element discretizations to finite difference and finite element discretizations addressing flow problems in very complicated geometries. A general iterative relaxation procedure (Zanolli patching) is employed that enforcesC ¹ continuity along the patching interface between the two differently discretized subdomains. In fluid flow simulations of transitional and turbulent flows the high-order discretization (spectral element) is used in the outer part of the domain where the Reynolds number is effectively very high. Near rough wall boundaries (where the flow is effectively very viscous) the use of low-order discretizations provides sufficient accuracy and allows for efficient treatment of the complex geometry. An analysis of the patching procedure is presented for elliptic problems, and extensions to incompressible Navier-Stokes equations are implemented using an efficient high-order splitting scheme. Several examples are given for elliptic and flow model problems and performance is measured on both serial and parallel processors.     

Iterative methods for domain decomposition using spectral tau and collocation approximation

In this paper we consider the numerical approximation of elliptic problems by spectral methods in domains subdivided into substructures. We review an iterative procedure with interface relaxation, reducing the given differential problem to a sequence of Dirichlet and mixed Neumann-Dirichlet problems on each subdomain. The iterative procedure is applied to both tau and collocation spectral approximations. Two optimal strategies for the automatic selection of the relaxation parameter are indicated. We present several numerical experiments showing the convergence properties of the iterative scheme with respect to the decomposition. A multilevel technique for domain decomposition methods is proposed.     

Gridless spectral algorithm for Stokes flow with moving boundaries

A gridless, spectrally-accurate algorithm for the Stokes flow with moving boundaries is presented. The algorithm uses fixed computational domain with boundaries of the flow domain moving inside the computational domain. The spatial discretization is based on the Fourier expansions in the streamwise direction and the Chebyshev expansions in the transverse direction. Temporal discretization uses one- and two-steps implicit formulations. The boundary conditions on the moving boundaries are imposed using the immersed boundary conditions concept. Numerical tests confirm the spectral accuracy in space and theoretically-predicted accuracy in time. Different variants of the solution procedure are presented and their relative advantages are discussed.     

From h to p efficiently: Strategy selection for operator evaluation on hexahedral and tetrahedral elements

A spectral/hp element discretisation permits both geometric flexibility and beneficial convergence properties to be attained simultaneously. The choice of elemental polynomial order has a profound effect on the efficiency of different implementation strategies with their performance varying substantially for low and high order spectral/hp discretisations. We examine how careful selection of the strategy minimises computational cost across a range of polynomial orders in three dimensions and compare how different operators, and the choice of element shape, lead to different break-even points between the implementations. In three dimensions, higher expansion orders quickly lead to a large increase in the number of element-interior modes, particularly in hexahedral elements. For a typical boundary–interior modal decomposition, this can rapidly lead to a poor performance from a global approach, while a sum-factorisation technique, exploiting the tensor-product structure of elemental expansions, leads to better performance. Furthermore, increased memory requirements may cause an implementation to show poor runtime performance on a given system, even if the strict operation count is minimal, due to detrimental caching effects and other machine-dependent factors.     

Spectral domain decomposition techniques for viscous incompressible flows

Spectral domain decomposition methods are described for solving the equations governing the flow of viscous incompressible fluids in rectangularly decomposable domains. The domain of interest is divided into a number of rectangular subdomains on each of which a spectral approximation of the flow variables is sought. For Newtonian flows a stream function formulation is used whereas for nonNewtonian flows the components of the extra-stress tensor are also used. Efficient direct methods for the solution of the algebraic systems are discussed. Numerical results are presented for laminar flow through a channel contraction and for the stick-slip problem.     

A Spectral Element Method to Price European Options. I. Single Asset with and without Jump Diffusion

We develop a spectral element method to price single factor European options with and without jump diffusion. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a Gauss-Lobatto mesh within each element, which allows an exact representation of the non-smooth payoff function. The convolution integral is approximated by high order Gauss-Lobatto quadratures. A second order implicit/explicit (IMEX) approximation is used to integrate in time, with the convolution integral integrated explicitly. The method is spectrally accurate (exponentially convergent) in space for the solution and Greeks, and second-order accurate in time. The spectral element solution to the Black-Scholes equation is ten to one hundred times faster than commonly used second order finite difference methods.     

基于分层模型的复合材料耦合Lamb波频率—波数域特性研究

针对复合材料层合板中耦合Lamb波的传播问题,基于分层模型提出解析建模与有限元数值模拟相结合的方法对其进行预测和评估。利用Legendre正交多项式展开法推导多层各向异性复合材料层合板中耦合Lamb波的控制方程,并对频率-波数域频散特性曲线实现数值求解。基于平面壳单元构建复合材料层合板的有限元模型,采用波结构加载法生成单一Lamb波基本模态,设计复合材料层合板的不同纤维取向、边界和界面约束条件,并经二维傅里叶变换获得有限元模拟数据的频率-波数域频散特性曲线。通过对比验证,结果表明两种方法均有较好的吻合性。     

Theoretical analysis of some spectral multigrid methods

We develop a theoretical analysis of a multigrid algorithm applied to spectral element discretization of linear elliptic problems. For a 1-D problem with non-constant coefficients we prove essentially the independence of the two-level convergence factor with respect to both the degree of the polynomial approximation and the number of spectral elements. We also sketch some ideas for the analysis of the 2-D case when only one spectral element is involved.     

A Hybrid Fourier–Chebyshev Method for Partial Differential Equations

We propose a pseudospectral hybrid algorithm to approximate the solution of partial differential equations (PDEs) with non-periodic boundary conditions. Most of the approximations are computed using Fourier expansions that can be efficiently obtained by fast Fourier transforms. To avoid the Gibbs phenomenon, super-Gaussian window functions are used in physical space. Near the boundaries, we use local polynomial approximations to correct the solution. We analyze the accuracy and eigenvalue stability of the method for several PDEs. The method compares favorably to traditional spectral methods, and numerical results indicate that for hyperbolic problems a time step restriction of O(1/N) is sufficient for stability. R.B. Platte’s address after December 2009: Arizona State University, Department of Mathematics and Statistics, Tempe, AZ, 85287-1804.     

Ray-tracing polymorphic multidomain spectral/hp elements for isosurface rendering

The purpose of this paper is to present a ray-tracing isosurface rendering algorithm for spectral/hp (high-order finite) element methods in which the visualization error is both quantified and minimized. Determination of the ray-isosurface intersection is accomplished by classic polynomial root-finding applied to a polynomial approximation obtained by projecting the finite element solution over element-partitioned segments along the ray. Combining the smoothness properties of spectral/hp elements with classic orthogonal polynomial approximation theory, we devise an adaptive scheme which allows the polynomial approximation along a ray-segment to be arbitrarily close to the true solution. The resulting images converge toward a pixel-exact image at a rate far faster than sampling the spectral/hp element solution and applying classic low-order visualization techniques such as marching cubes.     

Spectral element-Fourier methods for incompressible turbulent flows

A mixed spectral element-(Fourier) spectral method is proposed for solution of the incompressible Navier-Stokes equations in general, curvilinear domains. The formulation is appropriate for simulations of turbulent flows in complex geometries with only one homogeneous flow direction. The governing equations are written in a form suitable for both direct (DNS) and large-eddy (LES) simulations allowing a unified implementation. The method is based on skew-symmetric convective operators that induce minimal aliasing errors and fast Helmholtz solvers that employ efficient iterative algorithms (e.g. multigrid). Direct numerical simulations of channel flow verified that the proposed method can sustain turbulent fluctuations even at ‘marginal’ Reynolds numbers. The flexibility of the method to efficiently simulate complex-geometry flows is demonstrated through an example of transitional flow in a grooved channel and an example of transitional-turbulent flow over rough wall surfaces.     

A spectrally formulated plate element for wave propagation analysis in anisotropic material

A new spectrally formulated plate element is developed to study wave propagation in composite structures. The element is based on the classical lamination plate theory. Recently developed method based on singular value decomposition (SVD) is used in the element formulation. Along with this, a new strategy based on the method of solving polynomial eigenvalue problem (PEP) is proposed in this paper, which significantly reduces human intervention (and thus human error), in the element formulation. The developed element has an exact dynamic stiffness matrix, as it uses the exact solution of the governing elastodynamic equation of plate in frequency–wavenumber domain as the interpolating functions. Due to this, the mass distribution is modeled exactly, and as a result, a single element captures the exact frequency response of a regular structure, and it suffices to model a plate of any dimension. Thus, the cost of computation is dramatically reduced compared to the cost of conventional finite element analysis. The fast Fourier transform (FFT) and Fourier series are used for inversion to time–space domain. This element is used to model plate with ply drops and to capture the propagation of Lamb waves.     

MEMS stochastic model order reduction method based on polynomial chaos expansion

Modeling and simulation of MEMS devices is a very complex task which involve the electrical, mechanical, fluidic and thermal domains, and there are still some uncertainties need to be accounted because of uncertain material and/or geometric parameters factors. According to these problems, we put forward to stochastic model order reduction method under random input conditions to facilitate fast time and frequency domain analyses, the method firstly process model order reduction by Structure Preserving Reduced-order Interconnect Macro Modeling method, then makes use of polynomial chaos expansions in terms of the random input and output variables for the matrices of a finite element model of the system; at last we give the expected values and standard deviations computing method to MEMS stochastic model. The simulation results verify the method is effective in large scale MEMS design process.     

Scalable parallel FFT for spectral simulations on a Beowulf cluster

The implementation and performance of the multidimensional Fast Fourier Transform (FFT) on a distributed memory Beowulf cluster is examined. We focus on the three-dimensional (3D) real transform, an essential computational component of Galerkin and pseudo-spectral codes. The approach studied is a 1D domain decomposition algorithm that relies on communication-intensive transpose operation involving P processors. Communication is based upon the standard portable message passing interface (MPI). We show that 1/P scaling for execution time at fixed problem size N³ (i.e., linear speedup) can be obtained provided that (1) the transpose algorithm is optimized for simultaneous block communication by all processors; and (2) communication is arranged for non-overlapping pairwise communication between processors, thus eliminating blocking when standard fast ethernet interconnects are employed. This method provides the basis for implementation of scalable and efficient spectral method computations of hydrodynamic and magneto-hydrodynamic turbulence on Beowulf clusters assembled from standard commodity components. An example is presented using a 3D passive scalar code.     

Simulations of Time-Dependent Flows of Viscoelastic Fluids with Spectral Element Methods

This paper presents the application of spectral element methods to simulate the time-dependent flow of viscoelastic fluids in non-trivial geometries using a closed-form differential constitutive equation. As an example, results relative to the flow of a FENE-CR fluid in a two-dimensional four-to-one contraction are given.     

Domain Decomposition Spectral Method for Mixed Inhomogeneous Boundary Value Problems of High Order Differential Equations on Unbounded Domains

In this paper, we develop domain decomposition spectral method for mixed inhomogeneous boundary value problems of high order differential equations defined on unbounded domains. We introduce an orthogonal family of new generalized Laguerre functions, with the weight function x ^?? , ?? being any real number. The corresponding quasi-orthogonal approximation and Gauss-Radau type interpolation are investigated, which play important roles in the related spectral and collocation methods. As examples of applications, we propose the domain decomposition spectral methods for two fourth order problems, and the spectral method with essential imposition of boundary conditions. The spectral accuracy is proved. Numerical results demonstrate the effectiveness of suggested algorithms.     

A pseudo-spectral method with volume penalisation for magnetohydrodynamic turbulence in confined domains

We present a Fourier pseudo-spectral method for solving the resistive magnetohydrodynamic equations of incompressible flow in confined domains. A volume penalisation method allows to take into account boundary conditions and the geometry of the domain. A code validation is presented for the z-pinch test case. Numerical simulations of decaying MHD turbulence in two space dimensions show spontaneous spin-up of the flow in non-axisymmetric geometries, which is reflected by the generation of angular momentum. First results of decaying MHD turbulence in a cylinder illustrate the potential of the new method for three-dimensional simulations.     

Evaluation of integrals for a ten-node isoparametric tetrahedral finite element

The use of isoparametric finite elemts in solving three-dimensional problems typically requires the numerical evaluation of a large number of integrals over individual element domains. The evaluation of these integrals by numerical quadrature, which is the traditional approach, can be computationally expensive. For certain problems the present study provides a more efficient method for evaluation of the needed integrals. For these problems some or all of the desired integrals can be evaluated as linear combinations of basic integrals whose integrands are either (i) products of shape (interpolation) functions or (ii) a derivative of a shape function times a product of one or more shape functions. Basic integrals of these two types (when written in terms of local coordinate systems) have integrands which are polynomial both in the variables of integration and in the nodal coordinates and, thus, can be expressed as linear combinations (with rational number coefficients) of a set of polynomial functions of the nodal coordinates. Group theoretic techniques can be employed to select appropriate sets of polynomial functions for use in these expansions and to reduce substantially the number of basic integrals which need to be explicitly evaluated.

The details for the approach have been worked out for a ten-node isoparametric tetrahedral element through the use of MACSYMA, a computer system for algebraic manipulation. The symmetry group for this element has order 24. The basic integrals of types (i) and (ii) are expressed as linear combinations of 20 and 26 terms, respectively. The special case of a straight-edged tetrahedral element with mid-edge nodes is also discussed.