A Least-Squares Spectral Element Formulation for the Stokes Problem

Least-squares spectral element methods seem very promising since they combine the generality of finite element methods with the accuracy of the spectral methods and also the theoretical and computational advantages in the algorithmic design and implementation of the least-squares methods. The new element in this work is the choice of spectral elements for the discretization of the least-squares formulation for its superior accuracy due to the high-order basis-functions. The main issue of this paper is the derivation of a least-squares spectral element formulation for the Stokes equations and the role of the boundary conditions on the coercivity relations. The numerical simulations confirm the usual exponential rate of convergence when p-refinement is applied which is typical for spectral element discretization.     

Spectral element spatial discretization error in solving highly anisotropic heat conduction equation

This paper describes a study of the effects of the overall spatial resolution, polynomial degree and computational grid directionality on the accuracy of numerical solutions of a highly anisotropic thermal diffusion equation using the spectral element spatial discretization method. The high-order spectral element macroscopic modeling code SEL/HiFi has been used to explore the parameter space. It is shown that for a given number of spatial degrees of freedom, increasing polynomial degree while reducing the number of elements results in exponential reduction of the numerical error. The alignment of the grid with the direction of anisotropy is shown to further improve the accuracy of the solution. These effects are qualitatively explained and numerically quantified in 2- and 3-dimensional calculations with straight and curved anisotropy.     

Surface remeshing with robust high-order reconstruction

Remeshing is an important problem in variety of applications, such as finite element methods and geometry processing. Surface remeshing poses some unique challenges, as it must deliver not only good mesh quality but also good geometric accuracy. For applications such as finite elements with high-order elements (quadratic or cubic elements), the geometry must be preserved to high-order (third-order or higher) accuracy, since low-order accuracy may undermine the convergence of numerical computations. The problem is particularly challenging if the CAD model is not available for the underlying geometry, and is even more so if the surface meshes contain some inverted elements. We describe remeshing strategies that can simultaneously produce high-quality triangular meshes, untangling mildly folded triangles and preserve the geometry to high-order of accuracy. Our approach extends our earlier works on high-order surface reconstruction and mesh optimization by enhancing its robustness with a geometric limiter for under-resolved geometries. We also integrate high-order surface reconstruction with surface mesh adaptation techniques, which alter the number of triangles and nodes. We demonstrate the utilization of our method to meshes for high-order finite elements, biomedical image-based surface meshes, and complex interface meshes in fluid simulations.     

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.     

Finite element response sensitivity analysis: a comparison between force-based and displacement-based frame element models

This paper focuses on a comparison between displacement-based and force-based elements for static and dynamic response sensitivity analysis of frame type structures. Previous research has shown that force-based frame elements are superior to classical displacement-based elements enabling, at no significant additional computational costs, a drastic reduction in the number of elements required for a given level of accuracy in the simulated response. The present work shows that this advantage of force-based over displacement-based elements is even more conspicuous in the context of gradient-based optimization methods, which are used in several structural engineering sub-fields (e.g., structural optimization, structural reliability analysis, finite element model updating) and which require accurate and efficient computation of structural response and response sensitivities to material and loading parameters. The two methodologies for displacement-based and force-based element sensitivity computations are compared. Three application examples are presented to illustrate the conclusions. Material-only non-linearity is considered. Significant benefits are found in using force-based frame element models for both response and response sensitivity analysis in terms of trade-off between accuracy and computational cost.     

Identification of multiple flaws in 2D structures using dynamic extended spectral finite element method with a universally enhanced meta-heuristic optimizer

In this paper, flaw detection of two-dimensional structures is carried out using the extended spectral finite element method (XSFEM) associated with particle swarm optimization (PSO) algorithm enhanced by a new so-called active/inactive flaw (AIF) strategy. The AIF strategy, which is inspired from earthquake engineering concepts, is proposed for the first time in this paper. The XSFEM is employed to model the cracked and holed structures, while the PSO, which is a suitable non-gradient method for solving such problems, is employed to find crack location as an optimizer. The XSFEM consists of remarkable capabilities with the main features of spectral finite element method (SFEM) and extended finite element method (XFEM) to analyze the damaged structures without remeshing, leading it to be a proper approach in iterative processes. Moreover, the XSFEM enhances the accuracy of wave propagation analysis, and decreases computational cost as well in comparison with the XFEM. The application of XSFEM in damage detection of structures is studied for the first time in this paper. Furthermore, the AIF strategy is proposed in order to handle a simultaneously discrete and continuous optimization in an efficient way reducing computational effort. Considering the AIF as a universal strategy, it can be used in any meta-heuristic optimizer. In this research, the PSO is seeking for geometrical properties and the number of flaws in order to detect them by minimizing an error function based on sensor measurements. To overcome the challenge of unknown number of flaws, the proposed AIF strategy is employed in the PSO. Several benchmark examples are examined to evaluate capability and accuracy of the proposed algorithm for detection of cracks and holes.     

A high-order Lagrangian-decoupling method for the incompressible Navier-Stokes equations

In this paper we present a high-order Lagrangian-decoupling method for the unsteady convection diffusion and incompressible Navier-Stokes equations. The method is based upon Lagrangian variational forms that reduce the convection-diffusion equation to a symmetric initial value problem, implicit high-order backward-differentiation finite difference schemes for integration along characteristics, finite element or spectral element spatial discretizations and mesh-invariance procedures and high-order explicit time-stepping schemes for deducing function values at convected space-time points. The method improves upon previous finite element characteristic methods through the systematic and efficient extension to high-order accuracy and the introduction of a simple structure-preserving characteristic-foot calculation procedure which is readily implemented on modern architectures. The new method is significantly more efficient than explicit-convection schemes for the Navier-Stokes equations due to the decoupling of the convection and Stokes operators and the attendant increase in temporal stability. Numerous numerical examples are given for the convection-diffusion and Navier-Stokes equations for the particular case of a spectral element spatial discretization.     

Efficient Implementation of Smoothness-Increasing Accuracy-Conserving (SIAC) Filters for Discontinuous Galerkin Solutions

The discontinuous Galerkin (DG) methods provide a high-order extension of the finite volume method in much the same way as high-order or spectral/hp elements extend standard finite elements. However, lack of inter-element continuity is often contrary to the smoothness assumptions upon which many post-processing algorithms such as those used in visualization are based. Smoothness-increasing accuracy-conserving (SIAC) filters were proposed as a means of ameliorating the challenges introduced by the lack of regularity at element interfaces by eliminating the discontinuity between elements in a way that is consistent with the DG methodology; in particular, high-order accuracy is preserved and in many cases increased. The goal of this paper is to explicitly define the steps to efficient computation of this filtering technique as applied to both structured triangular and quadrilateral meshes. Furthermore, as the SIAC filter is a good candidate for parallelization, we provide, for the first time, results that confirm anticipated performance scaling when parallelized on a shared-memory multi-processor machine.     

Performance of the BEM solution in 3D acoustic wave scattering

A fixed cylindrical circular cavity and a cylindrical circular column of fluid of infinite length submerged in a homogeneous fluid medium, and subjected to a pressure point source, for which closed form solutions are known, are used to assess the performance of constant, linear and quadratic boundary elements in the analysis of acoustic scattering.This aim is accomplished by evaluating the error committed by the boundary element method (BEM) for a wide range of frequencies and wave numbers. First, the position of dominant BEM errors in the frequency versus spatial wave number domains are identified and related to the natural modes of vibration of the cylindrical circular inclusion. Then, the errors that occur by using constant, linear and quadratic elements are compared when the inclusion is modelled with the same number of nodes (i.e. maintaining computational cost). Finally, the importance of the position of the nodal points inside discontinuous boundary elements is analysed.     

基于非截断小波有限元的BLT正向问题研究

针对自发荧光断层成像,提出了一种非截断小波有限元算法.该算法采用单元间非截断组合小波基来逼近未知函数,从理论上解决了二维和三维下复杂形状体的剖分,并成功地应用于自发荧光断层成像正向问题中圆柱和圆球仿体的研究.理论分析和数值仿真结果表明,与传统有限元的数值解相比,该算法在获得同样有效解的情况下减少了单元剖分数,降低了计算的复杂度.     

A performance study of plane wave finite element methods with a Padé-type artificial boundary condition in acoustic scattering

This paper proposes an original numerical method and studies its performance for solving high-frequency scattering problems involving elongated scatterers. The approach is based on coupling a high-order Padé-type non-reflecting boundary condition with plane wave finite element formulations. It is shown that for some numerical examples the approximate solution of suitable accuracy can be obtained using a small number of degrees of freedom.     

A stabilized MITC element for accurate wave response in Reissner–Mindlin plates

Residual based finite element methods are developed for accurate time-harmonic wave response of the Reissner–Mindlin plate model. The methods are obtained by appending a generalized least-squares term to the mixed variational form for the finite element approximation. Through judicious selection of the design parameters inherent in the least-squares modification, this formulation provides a consistent and general framework for enhancing the wave accuracy of mixed plate elements. In this paper, the mixed interpolation technique of the well-established MITC4 element is used to develop a new mixed least-squares (MLS4) four-node quadrilateral plate element with improved wave accuracy. Complex wave number dispersion analysis is used to design optimal mesh parameters, which for a given wave angle, match both propagating and evanescent analytical wave numbers for Reissner–Mindlin plates. Numerical results demonstrates the significantly improved accuracy of the new MLS4 plate element compared to the underlying MITC4 element.     

Parallel implementation of finite-element/newton method for solution of steady-state and transient nonlinear partial differential equations

Domain decomposition by nested dissection for concurrent factorization and storage (CFS) of asymmetric matrices is coupled with finite element and spectral element discretizations and with Newton's method to yield an algorithm for parallel solution of nonlinear initial-and boundary-value problem. The efficiency of the CFS algorithm implemented on a MIMD computer is demonstrated by analysis of the solution of the two-dimensional, Poisson equation discretized using both finite and spectral elements. Computation rates and speedups for the LU-decomposition algorithm, which is the most time consuming portion of the solution algorithm, scale with the number of processors. The spectral element discretization with high-order interpolating polynomials yields especially high speedups because the ratio of communication to computation is lower than for low-order finite element discretizations. The robustness of the parallel implementation of the finite-element/Newton algorithm is demonstrated by solution of steady and transient natural convection in a two-dimensional cavity, a standard test problem for low Prandtl number convection. Time integration is performed using a fully implicit algorithm with a modified Newton's method for solution of nonlinear equations at each time step. The efficiency of the CFS version of the finite-element/Newton algorithm compares well with a spectral element algorithm implemented on a MIMD computer using iterative matrix methods.Submitted toJ. Scientific Computing, August 25, 1994.     

Optimized High-Order Derivative and Dissipation Operators Satisfying Summation by Parts,and Applications in Three-dimensional Multi-block Evolutions

We construct optimized high-order finite differencing operators which satisfy summation by parts. Since these operators are not uniquely defined, we consider several optimization criteria: minimizing the bandwidth, the truncation error on the boundary points, the spectral radius, or a combination of these. We examine in detail a set of operators that are up to tenth order accurate in the interior, and we surprisingly find that a combination of these optimizations can improve the operators’ spectral radius and accuracy by orders of magnitude in certain cases. We also construct high-order dissipation operators that are compatible with these new finite difference operators and which are semi-definite with respect to the appropriate summation by parts scalar product. We test the stability and accuracy of these new difference and dissipation operators by evolving a three-dimensional scalar wave equation on a spherical domain consisting of seven blocks, each discretized with a structured grid, and connected through penalty boundary conditions. In particular, we find that the constructed dissipation operators are effective in suppressing instabilities that are sometimes otherwise present in the restricted full norm case.     

Rosenbrock-type methods applied to finite element computations within finite strain viscoelasticity

In this article time-adaptive high-order Rosenbrock-type methods are applied to the system of differential–algebraic equations which results from the space-discretization using finite elements based on a constitutive model of finite strain viscoelasticity. It is shown that in this smooth problem more efficient finite element computations result in comparison to classical finite element approaches since the time integration on the basis of Rosenbrock-type methods does not lead to a system of non-linear equations. In other words, all aspects of implicit finite elements as local iterations on Gauss-point level and global equilibrium iterations do not occur. The first introduction to this approach proposed by Hartmann and Wensch [22] is extended here to the case of finite strain applications, where the geometrical non-linear deformation has an essential contribution to the non-linearities. Additionally, a clear decomposition into local (element or Gauss-point) work and global computational work using the Schur-complement is introduced to exploit the classical finite element character. Moreover, the extension to the reaction force computation, which is different to the classical approach, and the influence to mixed element formulations, here, the three-field formulation for displacements, pressure and dilatation, are discussed. The performance of various Rosenbrock-type methods is investigated and shows that for low accuracy requirements as in order one methods, the proposal yields a drastic reduction of the computational time.     

Optimized meshfree methods for acoustics

When the Helmholtz equation is solved by numerical methods as, e.g., the finite element method (FEM), the solution suffers from the so-called pollution effect. The pollution is mainly caused by the dispersion, meaning that the numerical wave number disagrees with the wave number of the exact solution. This leads to inaccurate results, especially for high wave numbers. In order to obtain acceptable results also for higher wave numbers, either a high element resolution or elements of a higher order can be used. For either option the consequence is an increased computation time and memory capacity.Meshfree methods as the element free Galerkin method (EFGM) and the radial point interpolation method (RPIM) are not dispersion-free either, but it has been shown that meshfree methods are able to reduce the dispersion significantly. Both methods offer several parameters, which can be modified in order to obtain optimal results with respect to the dispersion effect. This work presents an exhaustive parameter study on both the EFGM and the RPIM. It is shown, that the methods can be significantly improved if certain parameters as, e.g., weighting functions, shape parameters, size of the influence domain, are chosen appropriately.     

Error control and mesh optimization for high-order finite element approximation of incompressible viscous flow

We discuss the use of a posteriori error estimates for high-order finite element methods during simulation of the flow of incompressible viscous fluids. The correlation between the error estimator and actual error is used as a criterion for the error analysis efficiency. We show how to use the error estimator for mesh optimization which improves computational efficiency for both steady-state and unsteady flows. The method is applied to two-dimensional problems with known analytical solutions (Jeffrey-Hamel flow) and more complex flows around a body, both in a channel and in an open domain.     

一类二维粘性波动方程的交替方向有限体积元方法

针对二维粘性波动方程模型问题,提出了一类基于双线性插值的交替方向有限体积元方法,并给出了两种具体计算格式,一是基于有限差分方法中Douglas思想的格式,二是一类推广型的局部一维格式.分析证明了该方法按照L~2范数在时间和空间方向均有二阶收敛精度.最后,数值算例验证了算法的有效性和精确性.     

PyFR: An open source framework for solving advection–diffusion type problems on streaming architectures using the flux reconstruction approach

High-order numerical methods for unstructured grids combine the superior accuracy of high-order spectral or finite difference methods with the geometric flexibility of low-order finite volume or finite element schemes. The Flux Reconstruction (FR) approach unifies various high-order schemes for unstructured grids within a single framework. Additionally, the FR approach exhibits a significant degree of element locality, and is thus able to run efficiently on modern streaming architectures, such as Graphical Processing Units (GPUs). The aforementioned properties of FR mean it offers a promising route to performing affordable, and hence industrially relevant, scale-resolving simulations of hitherto intractable unsteady flows within the vicinity of real-world engineering geometries. In this paper we present PyFR, an open-source Python based framework for solving advection–diffusion type problems on streaming architectures using the FR approach. The framework is designed to solve a range of governing systems on mixed unstructured grids containing various element types. It is also designed to target a range of hardware platforms via use of an in-built domain specific language based on the Mako templating engine. The current release of PyFR is able to solve the compressible Euler and Navier–Stokes equations on grids of quadrilateral and triangular elements in two dimensions, and hexahedral elements in three dimensions, targeting clusters of CPUs, and NVIDIA GPUs. Results are presented for various benchmark flow problems, single-node performance is discussed, and scalability of the code is demonstrated on up to 104 NVIDIA M2090 GPUs. The software is freely available under a 3-Clause New Style BSD license (see www.pyfr.org).