Multipatch discontinuous Galerkin isogeometric analysis of composite laminates

This paper is concerned with the isogeometric analysis (IGA) of composite laminates under cylindrical bending. Non-uniform rational B-splines (NURBS) are employed as basis functions for both geometric and computational implementations. In order to account for multiple domains, each lamina is modeled as a single NURBS patch. This multipatch representation corresponds to decomposition of the computational domain (composite laminate) into non-overlapping subdomains. As NURBS patches are discontinuous across their boundaries, a standard FEA-like procedure does not work for multipatch IGA; an additional numerical technique is required for coupling NURBS patches. Therefore, in this paper, one of the discontinuous Galerkin (DG) methods, namely symmetric interior penalty Galerkin formulation, is employed to allow for discontinuities. For numerical calculations, a composite laminate with stacking sequences $$0^{\circ }{/}90^{\circ }$$ and $$0^{\circ }{/}90^{\circ }{/}0^{\circ }$$, respectively, is adopted. The stresses are calculated along the thickness of the composite laminate, subjected to a sinusoidal load, and they are compared with the analytical solutions. It is shown that DG–IGA gives a better approximation in comparison with the standard IGA.     

An hp‐adaptive discontinuous Galerkin method for shallow water flows

An adaptive spectral/hp discontinuous Galerkin method for the two‐dimensional shallow water equations is presented. The model uses an orthogonal modal basis of arbitrary polynomial order p defined on unstructured, possibly non‐conforming, triangular elements for the spatial discretization. Based on a simple error indicator constructed by the solutions of approximation order p and p?1, we allow both for the mesh size, h, and polynomial approximation order to dynamically change during the simulation. For the h‐type refinement, the parent element is subdivided into four similar sibling elements. The time‐stepping is performed using a third‐order Runge–Kutta scheme. The performance of the hp‐adaptivity is illustrated for several test cases. It is found that for the case of smooth flows, p‐adaptivity is more efficient than h‐adaptivity with respect to degrees of freedom and computational time. Copyright © 2010 John Wiley & Sons, Ltd.     

A discontinuous Galerkin method for the Navier–Stokes equations

The foundations of a new discontinuous Galerkin method for simulating compressible viscous flows with shocks on standard unstructured grids are presented in this paper. The new method is based on a discontinuous Galerkin formulation both for the advective and the diffusive contributions. High‐order accuracy is achieved by using a recently developed hierarchical spectral basis. This basis is formed by combining Jacobi polynomials of high‐order weights written in a new co‐ordinate system. It retains a tensor‐product property, and provides accurate numerical quadrature. The formulation is conservative, and monotonicity is enforced by appropriately lowering the basis order and performing h‐refinement around discontinuities. Convergence results are shown for analytical two‐ and three‐dimensional solutions of diffusion and Navier–Stokes equations that demonstrate exponential convergence of the new method, even for highly distorted elements. Flow simulations for subsonic, transonic and supersonic flows are also presented that demonstrate discretization flexibility using hp‐type refinement. Unlike other high‐order methods, the new method uses standard finite volume grids consisting of arbitrary triangulizations and tetrahedrizations. Copyright © 1999 John Wiley & Sons, Ltd.     

基于等几何分析的结构优化设计研究进展

结构等几何分析是计算固体力学领域一种新兴的数值方法,致力于将CAD（计算机辅助设计）和CAE（计算机辅助工程）纳入到统一的数学表达框架。等几何分析紧密联系几何信息,采用相同的数学表达将几何精确建模、结构分析和设计过程结合,为结构优化设计提供了新的选择和机会。相比基于有限元的结构优化方法,等几何优化设计方法可在一定程度上提高结构优化的精度、效率和便利性。本文针对具有代表性的结构等几何优化设计,包括形状优化、尺寸优化和拓扑优化等问题,系统梳理和综述了主要的等几何优化方法及其在结构优化设计中的应用。比较分析和评述了结构等几何优化设计方法的算法特点及计算优势与劣势,探讨了基于等几何分析的结构优化研究的前沿问题,并展望了未来的发展方向,包括：基于复杂剪裁CAD几何的高效等几何分析与优化设计、基于实体几何构造的结构等几何分析和优化设计、等几何分析与其他力学分析方法结合的结构优化、基于等几何分析的壳体优化设计、基于等几何分析的材料和结构一体化优化设计以及考虑不确定性的结构等几何优化设计等。     

Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory

In this paper, we present an effectively numerical approach based on isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT) for geometrically nonlinear analysis of laminated composite plates. The HSDT allows us to approximate displacement field that ensures by itself the realistic shear strain energy part without shear correction factors (SCFs). IGA utilizing basis functions namely B-splines or non-uniform rational B-splines (NURBS) enables to satisfy easily the stringent continuity requirement of the HSDT model without any additional variables. The nonlinearity of the plates is formed in the total Lagrange approach based on the small strain assumptions. Numerous numerical validations for the isotropic, orthotropic, cross-ply and angle-ply laminated plates are provided to demonstrate the effectiveness of the proposed method.     

Isogeometric analysis of free-form Timoshenko curved beams including the nonlinear effects of large deformations

In the present paper, the isogeometric analysis (IGA) of free-form planar curved beams is formulated based on the nonlinear Timoshenko beam theory to investigate the large deformation of beams with variable curvature. Based on the isoparametric concept, the shape functions of the field variables (displacement and rotation) in a finite element analysis are considered to be the same as the non-uniform rational basis spline (NURBS) basis functions defining the geometry. The validity of the presented formulation is tested in five case studies covering a wide range of engineering curved structures including from straight and constant curvature to variable curvature beams. The nonlinear deformation results obtained by the presented method are compared to well-established benchmark examples and also compared to the results of linear and nonlinear finite element analyses. As the nonlinear load-deflection behavior of Timoshenko beams is the main topic of this article, the results strongly show the applicability of the IGA method to the large deformation analysis of free-form curved beams. Finally, it is interesting to notice that, until very recently, the large deformations analysis of free-form Timoshenko curved beams has not been considered in IGA by researchers.     

Isogeometric analysis of nonlinear Euler–Bernoulli beam vibrations

In this paper we analyze the vibrations of nonlinear structures by means of the novel approach of isogeometric finite elements. The fundamental idea of isogeometric finite elements is to apply the same functions, namely B-Splines and NURBS (Non-Uniform Rational B-Splines), for describing the geometry and for representing the numerical solution. In case of linear vibrational analysis, this approach has already been shown to possess substantial advantages over classical finite elements, and we extend it here to a nonlinear framework based on the harmonic balance principle. As application, the straight nonlinear Euler–Bernoulli beam is used, and overall, it is demonstrated that isogeometric finite elements with B-Splines in combination with the harmonic balance method are a powerful means for the analysis of nonlinear structural vibrations. In particular, the smoother k-method provides higher accuracy than the p-method for isogeometric nonlinear vibration analysis.     

基于Bézier提取的三维等几何分析

等几何分析(IGA)将非均匀有理B样条(NURBS)函数作为有限元形函数,具有几何精确、高阶连续和精度高等优点。与常规有限元法C0连续的形函数不同,高阶IGA基函数不是定义在一个单元上,而是跨越由几个单元组成的参数空间,因而编程复杂且无法嵌入现有的有限元法计算框架及相应算法。本文建立了基于Bézier提取的三维IGA,将NURBS函数分解成伯恩斯坦多项式的线性组合,从而实现把NURBS单元分解为C0连续Bézier单元,这些单元与Lagrange单元相似,使IGA的实现和常规有限元一样,以便将IGA分析嵌入现有的有限元软件中。两个三维算例结果表明,基于Bézier提取的IGA和传统IGA的收敛性和精度相同。     

Generating an adaptive multiresolution image analysis with compact cupolets

We present an efficient control scheme for stabilizing unstable periodic orbits of chaotic systems. The resulting orbits are called cupolets and have been proven to be useful in the representation of oscillatory or quasi periodic signals such as appear in music and image compression (Short et al., AES 118th Convention preprint 6446, May 2005; Short et al., AES 119th Convention preprint 6588, October 2005). In this paper we show that these cupolets can be used effectively to produce an adaptive basis for the space of real-valued functions of a discrete variable. From this basis, we construct a multiresolution analysis which allows for the approximation of signals at different resolution levels and apply it to image compression. This adaptive multiresolution analysis provides an interesting continuum between Fourier analysis and wavelet analysis.     

Higher order spectral/hp finite element models of the Navier–Stokes equations

In this paper, we present higher order least-squares finite element formulations for viscous, incompressible, isothermal Navier–Stokes equations using spectral/hp basis functions. The second-order Navier–Stokes equations are recast as first-order system of equations using stresses as auxiliary variables. Both steady-state and transient problems are considered. For a better coupling of pressure and velocity, especially in transient flows, an iterative penalisation strategy is employed. The outflow-type boundary conditions are applied in a weak sense through the least-squares functional. The formulation is verified by solving various benchmark problems like the lid-driven cavity, backward-facing step and flow over cylinder problems using direct serial solver UMFPACK.     

Improved quadratic isogeometric element simulation of one-dimensional elastic wave propagation with central difference method

Two improved isogeometric quadratic elements and the central difference scheme are used to formulate the solution procedures of transient wave propagation problems. In the proposed procedures, the lumped matrices corresponding to the isogeometric elements are obtained. The stability conditions of the solution procedures are also acquired. The dispersion analysis is conducted to obtain the optimal Courant-Friedrichs-Lewy (CFL) number or time-step sizes corresponding to the spatial isogeometric elements. The dispersion analysis shows that the isogeometric quadratic element of the fourth-order dispersion error (called the isogeometric analysis (IGA)-f quadratic element) provides far more desirable numerical dissipation/dispersion than the element of the second-order dispersion error (called the IGA-s quadratic element) when appropriate time-step sizes are selected. The numerical simulations of one-dimensional (1D) transient wave propagation problems demonstrate the effectiveness of the proposed solution procedures.     

基于IGA-SIMP法的连续体结构应力约束拓扑优化

建立了一种IGA-SIMP框架下的连续体结构应力约束拓扑优化方法。基于常用的SIMP模型,将非均匀有理B样条（NURBS）函数用于几何建模、结构分析和设计参数化,实现了结构分析和优化设计的集成统一。利用高阶连续的NURBS基函数,等几何分析（IGA）提高了结构应力及其灵敏度的计算精度,增加了拓扑优化结果的可信性。为处理大量局部应力约束,提出了基于稳定转换法修正的P-norm应力约束策略,以克服拓扑优化中的迭代振荡和收敛困难。通过几个典型平面应力问题的拓扑优化算例表明了本文方法的有效性和精确性。应力约束下的体积最小化设计以及体积和应力约束下的柔顺度最小化设计的算例表明,基于稳定转换法修正的约束策略可以抑制应力约束体积最小化设计中的迭代振荡现象,获得稳定收敛的优化解;比较而言,体积和应力约束下的柔顺度最小化设计的迭代过程更加稳健,适合采用精确修正的应力约束策略。     

Bifurcation of Limit Cycles from a Polynomial Non-global Center

Consider the planar ordinary differential equation , where the set consists of k non-zero points. In this paper we perturb this vector field with a general polynomial perturbation of degree n and study how many limit cycles bifurcate from the period annulus of the origin in terms of k and n. One of the key points of our approach is that the Abelian integral that controls the bifurcation can be explicitly obtained as an application of the integral representation formula of harmonic functions through the Poisson kernel. Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

An hp‐adaptive unstructured finite volume solver for compressible flows

The idea of hp‐adaptation, which has originally been developed for compact schemes (such as finite element methods), suggests an adaptation scheme using a mixture of mesh refinement and order enrichment based on the smoothness of the solution to obtain an accurate solution efficiently. In this paper, we develop an hp‐adaptation framework for unstructured finite volume methods using residual‐based and adjoint‐based error indicators. For the residual‐based error indicator, we use a higher‐order discrete operator to estimate the truncation error, whereas this estimate is weighted by the solution of the discrete adjoint problem for an output of interest to form the adaptation indicator for adjoint‐based adaptations. We perform our adaptation by local subdivision of cells with nonconforming interfaces allowed and local reconstruction of higher‐order polynomials for solution approximations. We present our results for two‐dimensional compressible flow problems including subsonic inviscid, transonic inviscid, and subsonic laminar flow around the NACA 0012 airfoil and also turbulent flow over a flat plate. Our numerical results suggest the efficiency and accuracy advantages of adjoint‐based hp‐adaptations over uniform refinement and also over residual‐based adaptation for flows with and without singularities.     

High-order asymptotic analysis for the crack in nonlinear material

Accurate high-order asymptotic analyses were carried out for Mode II plane strain crack in power hardening materials. The second-order crack tip fields have been obtained. It is found that the amplitude coefficientk ₂ of the second term of the asymptotic field is correlated to the first order field as the hardening exponentn<n ^* (n ^*≈5), but asn≥n ^*,k ₂ turns to become an independent parameter. Our results also indicated that, the second term of the asymptotic field has little influence on the near-crack-tip field and can be neglected whenn<n ^*. In fact,k ₂ directly reflects the effects of triaxiality near the crack tip, the crack geometry and the loading mode, so that besidesJ-integral it can be used as another characteristic parameter in the two-parameter criterion. The project supported by National Natural Science Foundation of China     

Numerical investigation of inviscid shock wave dynamics in an expansion tube

Highly complicated shock wave dynamics has been numerically calculated by solving the Euler equations for a circular shock tube suddenly expanded three times of the original tube diameter atx=0. Shock waves of different shock Mach number,M _s =1.5 and 2.0, have produced remarkably distinct blast jet structures. A planar shock wave took its final form after the blast by repeated Mach reflections of the blast wave: the first one at the wall and the second one at the central axis. The central Mach disc overtook and merged with the annular Mach stem before the planar shock wave was formed. In contrast to the blast wave which would propagate spherically in an open space, the present blast wave undergoes complex morphological transformation in the restricted flow passage, resulting in an unstable and oscillatory blast jet structure of highly rotational nature. The slipstream tube emanating from the shock tube exit corner decomposed into a chain of small vortex rings that interacted with the barrel shock of the jet, which caused periodic collapse of the jet structure. The finite volume-FCT formulation equipped with the time-dependenth-refinement adaptive unstructured triangular mesh technique in the present paper has contributed to resolution of the intricate physical discontinuities developing in the blast flow fields.     

One dimensional lattice gases with rapidly decreasing interaction

We consider the pressure and the correlation functions of a one dimensional lattice gas in which the mutual interaction decreases as r exp-n ^t, (r, t>0), when the interparticle distance n. We prove that such a system cannot show phase transitions of order k1 in the sense that the pressure and the correlation functions are infinitely differentiable with respect to any relevant parameter (such as the temperature or the chemical potential).     

A note on the stability and accuracy of Cn finite elements in steady diffusion-convection problems

The paper is concerned with stability and accuracy of an nth order Lagrangian family of finite element steady-state solutions of the diffusion-convection equation, and furthermore is concerned with the stability and the accuracy of on mth kind Hermitian family of finite element solutions. We discuss the stability of the numerical solution based on the fact that the characteristic finite element solution can be expressed approximately as a rational function of cell Peclet number Pe_c ( = uh/k). Moreover, it is shown that by eliminating derivatives and by using the interpolation method over elements a stable solution is obtained over the domain independent of Pe_c for P^1,3, and for P^2,5 the stable solution is obtained for Pe_c less than 44.4.     

Higher‐order and adaptive discontinuous Galerkin methods with shock‐capturing applied to transonic turbulent delta wing flow

Discontinuous Galerkin (DG) methods allow high‐order flow solutions on unstructured or locally refined meshes by increasing the polynomial degree and using curved instead of straight‐sided elements. DG discretizations with higher polynomial degrees must, however, be stabilized in the vicinity of discontinuities of flow solutions such as shocks. In this article, we device a consistent shock‐capturing method for the Reynolds‐averaged Navier–Stokes and k‐ ω turbulence model equations based on an artificial viscosity term that depends on element residual terms. Furthermore, the DG method is combined with a residual‐based adaptation algorithm that targets at resolving all flow features. The higher‐order and adaptive DG method is applied to a fully turbulent transonic flow around the second Vortex Flow Experiment (VFE‐2) configuration with a good resolution of the vortex system.Copyright © 2013 John Wiley & Sons, Ltd.     

Adjoint‐based airfoil optimization with discretization error control

In this article, we develop a new airfoil shape optimization algorithm based on higher‐order adaptive DG methods with control of the discretization error. Each flow solution in the optimization loop is computed on a sequence of goal‐oriented h‐refined or hp‐refined meshes until the error estimation of the discretization error in a flow‐related target quantity (including the drag and lift coefficients) is below a prescribed tolerance. Discrete adjoint solutions are computed and employed for the multi‐target error estimation and adaptive mesh refinement. Furthermore, discrete adjoint solutions are employed for evaluating the gradients of the objective function used in the CGs optimization algorithm. Furthermore, an extension of the adjoint‐based gradient evaluation to the case of target lift flow computations is employed. The proposed algorithm is demonstrated on an inviscid transonic flow around the RAE2822, where the shape is optimized to minimize the drag at a given constant lift and airfoil thickness. The effect of the accuracy of the underlying flow solutions on the quality of the optimized airfoil shapes is investigated. Copyright © 2014 John Wiley & Sons, Ltd.