Generalized finite element method using proper orthogonal decomposition

A methodology is presented for generating enrichment functions in generalized finite element methods (GFEM) using experimental and/or simulated data. The approach is based on the proper orthogonal decomposition (POD) technique, which is used to generate low‐order representations of data that contain general information about the solution of partial differential equations. One of the main challenges in such enriched finite element methods is knowing how to choose, a priori, enrichment functions that capture the nature of the solution of the governing equations. POD produces low‐order subspaces, that are optimal in some norm, for approximating a given data set. For most problems, since the solution error in Galerkin methods is bounded by the error in the best approximation, it is expected that the optimal approximation properties of POD can be exploited to construct efficient enrichment functions. We demonstrate the potential of this approach through three numerical examples. Best‐approximation studies are conducted that reveal the advantages of using POD modes as enrichment functions in GFEM over a conventional POD basis. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite element analysis of internal flows with heat transfer

This paper presents a finite element-based model for the prediction of 2-D and 3-D internal flow problems. The Eulerian velocity correction method is used which can render a fast finite element code comparable with the finite difference methods. Nine different models for turbulent flows are incorporated in the code. A modified wall function approach for solving the energy equation with high Reynolds number models is presented for the first time. This is an extension of the wall function approach of Benim and Zinser and the method is insensitive to initial approximation. The performance of the nine turbulent models is evaluated by solving flow through pipes. The code is used to predict various internal flows such as flow in the diffuser and flow in a ribbed channel. The same Eulerian velocity correction method is extended to predict the 3-D laminar flows in various ducts. The steady state results have been compared with benchmark solutions and the agreement appears to be good.     

A Treatment of wall boundaries for turbulent flows by the use of a transmission finite element method

A method to connect momentum Navier-Stokes equations with the universal law of the wall using the finite element method is developed for turbulent wall flows. This method is based on a domain decomposition of the fluid into subdomains near a solid boundary where the law of the wall is valid. A transmission formulation is introduced to match these regions and a new class of boundary finite element is used. This finite element takes into account the near-wall profile of the velocity and the transmission conditions. Computational results are presented for Poiseuille flow and flow over a backward-facing step.     

Development of a scalable finite element solution to the Navier–Stokes equations

A scalable numerical model to solve the unsteady incompressible Navier–Stokes equations is developed using the Galerkin finite element method. The coupled equations are decoupled by the fractional-step method and the systems of equations are inverted by the Krylov subspace iterations. The data structure makes use of a domain decomposition of which each processor stores the parameters in its subdomain, while the linear equations solvers and matrices constructions are parallelized by a data parallel approach. The accuracy of the model is tested by modeling laminar flow inside a two-dimensional square lid-driven cavity for Reynolds numbers at 1,000 as well as three-dimensional turbulent plane and wavy Couette flow and heat transfer at high Reynolds numbers. The parallel performance of the code is assessed by measuring the CPU time taken on an IBM SP2 supercomputer. The speed up factor and parallel efficiency show a satisfactory computational performance.The authors wish to acknowledge Mr. W. K. Kwan of The University of Hong Kong for his help in using the IBM SP2 supercomputer.     

Structure and dynamics of low Reynolds number turbulent pipe flow

Using large-scale numerical calculations, we explore the proper orthogonal decomposition of low Reynolds number turbulent pipe flow, using both the translational invariant (Fourier) method and the method of snapshots. Each method has benefits and drawbacks, making the 'best' choice dependent on the purpose of the analysis. Owing to its construction, the Fourier method includes all the flow fields that are translational invariants of the simulated flow fields. Thus, the Fourier method converges to an estimate of the dimension of the chaotic attractor in less total simulation time than the method of snapshots. The converse is that for a given simulation, the method of snapshots yields a basis set that is more optimal because it does not include all of the translational invariants that were not a part of the simulation. Using the Fourier method yields smooth structures with definable subclasses based upon Fourier wavenumber pairs, and results in a new dynamical systems insight into turbulent pipe flow. These subclasses include a set of modes that propagate with a nearly constant phase speed, act together as a wave packet and transfer energy from streamwise rolls. It is these interactions that are responsible for bursting events and Reynolds stress generation. These structures and dynamics are similar to those found in turbulent channel flow. A comparison of structures and dynamics in turbulent pipe and channel flows is reported to emphasize the similarities and differences.     

Large eddy simulation of turbulent flows in external flow field using three-step FEM–BEM model

An innovative computational model is presented for the large eddy simulation (LES) modeling of multi-dimensional unsteady turbulent flow problems in external flow field. Based on the LES principles, the model uses a pressure projection method to solve the Navier–Stokes equations in transient condition. The turbulent motion is simulated by Smagorinsky sub-grid scale (SGS) eddy viscosity model. The momentum equation of the flow motion is solved using a three-step finite element method (FEM). The external flow field is simulated using a boundary element method (BEM) by solving a pressure Poisson equation that assumes the pressure as zero at the infinity. Through extracting the boundary effects on a specified finite computational domain, the model is able to solve the infinite boundary value problems. The present model is used to simulate the flows past a two-dimensional square rib and a three-dimensional cube at high Reynolds number. The simulation results are found to be reasonable and comparable with other models available in the literature even for coarse meshes.     

基于POD降阶方法的复合材料曲壁板颤振响应特性研究

建立了三维复合材料曲壁板的气动弹性有限元方程,将本征正交分解方法(POD)应用于三维复合材料曲壁板的非线性颤振响应降阶分析中,通过POD方法构造三维复合材料曲壁板颤振响应的POD模态,然后将系统的运动方程变换到POD模态坐标下,通过数值积分方法计算三维复合材料曲壁板的颤振响应,与传统的模态缩减法计算结果相比,结果很好的吻合,且大大节省了计算时间。     

Global and local POD models for the prediction of compressible flows with DG methods

Proper orthogonal decomposition (POD) allows to compress information by identifying the most energetic modes obtained from a database of snapshots. In this work, POD is used to predict the behavior of compressible flows by means of global and local approaches, which exploit some features of a discontinuous Galerkin spatial discretization. The presented global approach requires the definition of high‐order and low‐order POD bases, which are built from a database of high‐fidelity simulations. Predictions are obtained by performing a cheap low‐order simulation whose solution is projected on the low‐order basis. The projection coefficients are then used for the reconstruction with the high‐order basis. However, the nonlinear behavior related to the advection term of the governing equations makes the use of global POD bases quite problematic. For this reason, a second approach is presented in which an empirical POD basis is defined in each element of the mesh. This local approach is more intrusive with respect to the global approach but it is able to capture better the nonlinearities related to advection. The two approaches are tested and compared on the inviscid compressible flow around a gas‐turbine cascade and on the compressible turbulent flow around a wind turbine airfoil.     

hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks

A high‐order generalized finite element method (GFEM) for non‐planar three‐dimensional crack surfaces is presented. Discontinuous p‐hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non‐planar, non‐smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non‐planar curved crack fronts and the robustness of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

The generalized finite element method: an example of its implementation and illustration of its performance

The generalized finite element method (GFEM) was introduced in Reference 1 as a combination of the standard FEM and the partition of unity method. The standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data (the geometry of the domain, the loads, and the boundary conditions). The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation. The essential boundary conditions can be imposed exactly as in the standard FEM. Adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation. This paper gives an example of how the GFEM can be developed for the Laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the GFEM versus the standard FEM. Copyright © 2000 John Wiley & Sons, Ltd.     

A two-scale approach for the analysis of propagating three-dimensional fractures

This paper presents a generalized finite element method (GFEM) for crack growth simulations based on a two-scale decomposition of the solution—a smooth coarse-scale component and a singular fine-scale component. The smooth component is approximated by discretizations defined on coarse finite element meshes. The fine-scale component is approximated by the solution of local problems defined in neighborhoods of cracks. Boundary conditions for the local problems are provided by the available solution at a crack growth step. The methodology enables accurate modeling of 3-D propagating cracks on meshes with elements that are orders of magnitude larger than those required by the FEM. The coarse-scale mesh remains unchanged during the simulation. This, combined with the hierarchical nature of GFEM shape functions, allows the recycling of the factorization of the global stiffness matrix during a crack growth simulation. Numerical examples demonstrating the approximating properties of the proposed enrichment functions and the computational performance of the methodology are presented.     

Dimension reduction model for two-spatial dimensional stochastic wind field: Hybrid approach of spectral decomposition and wavenumber spectral representation

A renewed methodology for simulating two-spatial dimensional stochastic wind field is addressed in the present study. First, the concept of cross wavenumber spectral density (WSD) function is defined on the basis of power spectral density (PSD) function and spatial coherence function to characterize the spatial variability of the stochastic wind field in the two-spatial dimensions. Then, the hybrid approach of spectral representation and wavenumber spectral representation and that of proper orthogonal decomposition and wavenumber spectral representation are respectively derived from the Cholesky decomposition and eigen decomposition of the constructed WSD matrices. Immediately following that, the uniform hybrid expression of spectral decomposition and wavenumber spectral representation is obtained, which integrates the advantages of both the discrete and continuous methods of one-spatial dimensional stochastic field, allowing for reflecting the spatial characteristics of large-scale structures. Moreover, the dimension reduction model for two-spatial dimensional stochastic wind field is established via adopting random functions correlating the high-dimensional orthogonal random variables with merely 3 elementary random variables, such that this explicitly describes the probability information of stochastic wind field in probability density level. Finally, the numerical investigations of the two-spatial dimensional stochastic wind fields respectively acting on a long-span suspension bridge and a super high-rise building are implemented embedded in the FFT algorithm. The validity and engineering applicability of the proposed method are thus fully verified, providing a potentially effective approach for refined wind-resistance dynamic reliability analysis of large-scale complex engineering structures.     

微管道内湍流转捩的实验研究

研究旨在确定微管道内流动从层流到湍流转捩的临界雷诺数。利用微观粒子图像测速技术(Micro-PIV)研究了去离子水在内径为230μm的圆形截面玻璃微管道内的流场结构,得到了从层流到充分发展湍流各流动状态下的轴向平均速度分布和湍流度分布,实验雷诺数为1020~3145,同时研究了微管道内的流动阻力特性。平均速度场和脉动速度场的实验结果表明微管道内从层流到湍流的转捩发生在Re=1800~1900左右,与流动阻力的测量结果一致,与宏观流动比较,并未发现微管道内的流动转捩有明显提前。实验结果还显示,当Re>2700时,微管道内的平均流速分布和相对湍流度分布呈现典型的充分发展湍流状态特征。     

DIRECT FORMULATION FINITE ELEMENT METHOD FOR TWO-DIMENSIONAL GROUNDWATER POLLUTION MODELLING WITH A COMPARISON WITH CONVENTIONAL FINITE ELEMENT METHOD

Real world ground water pollution modelling deals with solute transport through anisotropic, heterogeneous media. The applicability of analytical solutions for such a real world system is extremely limited. As an effective tool, numerical models, such as finite difference and finite element methods, are usually employed to model field scenarios. Nevertheless, ground water pollution modelling is a hallenging task and frequently ends up with misleading results. Most of the time insufficient data are blamed for such erratic results. A recent investigation shows that the shortcomings of numerical formulations may be the major cause for many disputes and confusions in numerical analyses. In reality, a point injection of water in a static, homogeneous and isotropic groundwater system shows a radial dissipation of water forming a sphere; and a full-depth line injection shows a radial dissipation forming a cylinder. The finite difference method completely ignores this fundamental flow principles and allows water only to flow along orthogonal directions. To overcome this limitation, the finite element method was developed as a flexible approach in order to connect a node with the neighbouring nodes in various directions where water is assumed to flow in any directions along node connections. In a recent investigation, it has been found that the conventional finite element method does not keep the commitments; and its formulation techniques lead to a global matrix where a solution domain is not connected with all the neighbouring nodes and does not comply with the control-volume mass balance concept. A consistent finite element formulation approach which does not need imaginary mathematical formulation and overcomes the limitations of both the conventional finite difference and finite element methods has been developed. This method allows fluid flow and solute transport in a porous medium in radial directions. The global matrices for flow and transport obtained from this technique are field representative, diagonally dominant and easily convergent. The new method is robust, needs less mathematical computation and has many advantages over the conventional finite difference and finite element methods.     

The least-squares finite element method for low-mach-number compressible viscous flows

The present paper reports the development of the Least-Squares Finite Element Method (LSFEM) for simulating compressible viscous flows at low Mach numbers in which the incompressible flows pose as an extreme. The conventional approach requires special treatments for low-speed flows calculations: finite difference and finite volume methods are based on the use of the staggered grid or the preconditioning technique, and finite element methods rely on the mixed method and the operator-splitting method. In this paper, however, we show that such a difficulty does not exist for the LSFEM and no special treatment is needed. The LSFEM always leads to a symmetric, positive-definite matrix through which the compressible flow equations can be effectively solved. Two numerical examples are included to demonstrate the method: driven cavity flows at various Reynolds numbers and buoyancy-driven flows with significant density variation. Both examples are calculated by using full compressible flow equations.     

An extended/generalized phase-field finite element method for crack growth with global-local enrichment

An extended/generalized finite element method (XFEM/GFEM) for simulating quasistatic crack growth based on a phase-field method is presented. The method relies on approximations to solutions associated with two different scales: a global scale, that is, structural and discretized with a coarse mesh, and a local scale encapsulating the fractured region, that is, discretized with a fine mesh. A stable XFEM/GFEM is employed to embed the displacement and damage fields at the global scale. The proposed method accommodates approximation spaces that evolve between load steps, while preserving a fixed background mesh for the structural problem. In addition, a prediction-correction algorithm is employed to facilitate the dynamic evolution of the confined crack regions within a load step. Several numerical examples of benchmark problems in two- and three-dimensional quasistatic fracture are provided to demonstrate the approach.     

Application of proper orthogonal decomposition to the discrete Euler equations

The response of a fluid moving above a panel to localized oscillation of the panel is predicted using reduced‐order modelling (ROM) with the proper orthogonal decomposition technique. The flow is assumed to be inviscid and is modelled with the Euler equations. These non‐linear equations are discretized with a total‐variation diminishing algorithm and are projected onto an energy‐optimal subspace defined by an energy‐threshold criterion applied to a modal representation of time series data. Results are obtained for a bump oscillating in a Mach 1.2 flow. ROM is found to reduce the degrees of freedom necessary to simulate the flowfield by three orders of magnitude while preserving solution accuracy. Other observed benefits of ROM include increased allowable time step and robustness to variation of oscillation amplitude. Published in 2002 by John Wiley & Sons, Ltd.     

Numerical investigation of laminar separated flow through a channel with symmetric double expansion

Summary Two-dimensional Navier-Stokes equations have been solved numerically by a finite difference technique on a staggered grid for investigating the laminar flow of an incompressible viscous fluid through a channel with symmetric double expansion. A coordinate transformation has been employed to map the infinite irregular domain into a finite regular computational domain. At higher Reynolds numbers the flow becomes time-dependent and asymmetric. Such unsteadiness and asymmetry remains in the flow even up to turbulent flow conditions. The Pressure-Poisson equation has been solved and a pressure-velocity correction scheme has been invoked. Depending on the geometry and Reynolds number, secondary separation has been observed.     

A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach

The Galerkin finite element method (GFEM) owes its popularity to the local nature of nodal basis functions, i.e., the nodal basis function, when viewed globally, is non-zero only over a patch of elements connecting the node in question to its immediately neighboring nodes. The boundary element method (BEM), on the other hand, reduces the dimensionality of the problem by one, through involving the trial functions and their derivatives, only in the integrals over the global boundary of the domain; whereas, the GFEM involves the integration of the “energy” corresponding to the trial function over a patch of elements immediately surrounding the node. The GFEM leads to banded, sparse and symmetric matrices; the BEM based on the global boundary integral equation (GBIE) leads to full and unsymmetrical matrices. Because of the seemingly insurmountable difficulties associated with the automatic generation of element-meshes in GFEM, especially for 3-D problems, there has been a considerable interest in element free Galerkin methods (EFGM) in recent literature. However, the EFGMs still involve domain integrals over shadow elements and lead to difficulties in enforcing essential boundary conditions and in treating nonlinear problems. The object of the present paper is to present a new method that combines the advantageous features of all the three methods: GFEM, BEM and EFGM. It is a meshless method. It involves only boundary integration, however, over a local boundary centered at the node in question; it poses no difficulties in satisfying essential boundary conditions; it leads to banded and sparse system matrices; it uses the moving least squares (MLS) approximations. The method is based on a Local Boundary Integral Equation (LBIE) approach, which is quite general and easily applicable to nonlinear problems, and non-homogeneous domains. The concept of a “companion solution” is introduced so that the LBIE for the value of trial solution at the source point, inside the domain Ω of the given problem, involves only the trial function in the integral over the local boundary Ω _s of a sub-domain Ω _s centered at the node in question. This is in contrast to the traditional GBIE which involves the trial function as well as its gradient over the global boundary Γ of Ω. For source points that lie on Γ, the integrals over Ω _s involve, on the other hand, both the trial function and its gradient. It is shown that the satisfaction of the essential as well as natural boundary conditions is quite simple and algorithmically very efficient in the present LBIE approach. In the example problems dealing with Laplace and Poisson's equations, high rates of convergence for the Sobolev norms ||·||₀ and ||·||₁ have been found. In essence, the present EF-LBIE (Element Free-Local Boundary Integral Equation) approach is found to be a simple, efficient, and attractive alternative to the EFG methods that have been extensively popularized in recent literature.     

模型表面粗糙度对冷却塔风荷载的影响

采用两种粗糙条数量和五种粗糙条厚度分析模型表面粗糙度对冷却塔风荷载的影响,应用本征正交分解法(POD)进行风压点的加密和重构,分析不同粗糙度下冷却塔的风压分布和总体受力,并将试验结果与规范、以往的实测和风洞试验结果进行比较.研究发现:冷却塔模型的表面越光滑,喉部附近B层测点最人负风压的绝对值越大;当模型表面粗糙条数量增...