Computations of high‐speed compressible flows with adaptive cell‐centered finite element method

Abstract

An upwind cell‐centered finite element formulation is combined with an adaptive meshing technique to solve Navier‐Stokes equations for high‐speed inviscid and viscous compressible flows. The finite element formulation and the computational procedure are described. An adaptive meshing technique is applied to increase the analysis solution accuracy, as well as to minimize the computational time and the computer memory requirement. The efficiency of the combined method is evaluated by the examples of Mach 2.6 inviscid flow in a channel with compression and expansion ramps, Mach 6.47 inviscid and viscous flows past a cylinder, and Mach 4 viscous flow over a flat plate.     

Flux‐difference splitting scheme with modified multidimensional dissipation on unstructured meshes

Abstract

A flux‐difference splitting scheme with a modified multidimensional dissipation for high‐speed compressible flow analysis on unstructured meshes is presented. The scheme eliminates unphysical flow behaviors such as a spurious bump of the carbuncle phenomenon that occurs on the bow shock from flow over a blunt body, and the expansion shock generated from flow over a forward facing step. The switching function suggested by Quirk is implemented as a choice to detect the vicinity of strong shock. The proposed scheme is further extended to obtain higher‐order spatial and temporal solution accuracy. The scheme is, in addition, combined with an adaptive meshing technique that generates unstructured triangular meshes to resemble the flow phenomena for reducing computational effort. The entire procedure is evaluated by solving several benchmarks as well as steady‐state and transient high‐speed compressible flow problems.     

Refined h-adaptive finite element procedure for large deformation geotechnical problems

Adaptive finite element procedures automatically refine, coarsen, or relocate elements in a finite element mesh to obtain a solution with a specified accuracy. Although a significant amount of research has been devoted to adaptive finite element analysis, this method has not been widely applied to nonlinear geotechnical problems due to their complexity. In this paper, the h-adaptive finite element technique is employed to solve some complex geotechnical problems involving material nonlinearity and large deformations. The key components of h-adaptivity including robust mesh generation algorithms, error estimators and remapping procedures are discussed. This paper includes a brief literature review as well as formulation and implementation details of the h-adaptive technique. Finally, the method is used to solve some classical geotechnical problems and results are provided to illustrate the performance of the method.     

A point interpolation method for two‐dimensional solids

A point interpolation method (PIM) is presented for stress analysis for two‐dimensional solids. In the PIM, the problem domain is represented by properly scattered points. A technique is proposed to construct polynomial interpolants with delta function property based only on a group of arbitrarily distributed points. The PIM equations are then derived using variational principles. In the PIM, the essential boundary conditions can be implemented with ease as in the conventional finite element methods. The present PIM has been coded in FORTRAN. The validity and efficiency of the present PIM formulation are demonstrated through example problems. It is found that the present PIM is very easy to implement, and very flexible for obtained displacements and stresses of desired accuracy in solids. As the elements are not used for meshing the problem domain, the present PIM opens new avenues to develop adaptive analysis codes for stress analysis in solids and structures. Copyright © 2001 John Wiley & Sons, Ltd.     

Thermoelastic analysis of a layered slab subjected to a moving line heat source

Abstract

This paper reports a theoretical study of the transient thermal stress distributions of a layered slab which is composed of two different materials. The layered slab is heated by a moving line heat source on its upper surface and cooled convectively on the lower surface. In order to solve the initial and boundary value problem, a general hybrid Laplace transform/finite element method is utilized. Finally, a numerical procedure, the Fourier series technique, is used to obtain the inversion of the Laplace transform. The effect of the number of mesh elements in the X‐direction is also investigated to verify the accuracy and convergence of the finite element method. In addition, a typical result is compared with the analytic solution. The numerical results of the transient temperature and thermal stress distribution of the layered slab are presented to demonstrate the effect of the physical properties.     

Solidification in castings by finite element method

Abstract

A self-contained CAD (computer aided design) system capable of analyzing foundry casting processes in sand and gravity dies is being developed at the University College of Swansea. The work involves preprocessing, postprocessing, and a finite element code with some novel numerical techniques. The solidification of castings is a heat transfer problem involving phase change, which may occur in a narrow range of temperatures. To simulate the phenomena accurately, very fine meshes must be used and the solution of such a system becomes very expensive. In the Swansea system, an adaptive remeshing technique is introduced, which tracks the moving front of the phase change zone. At every time step, a scan is made to determine the points at which phase change is occurring, so that the remeshing may be done to produce a refined mesh at such points. The computing process is then continued. Examples have illustrated that the method is efficient and accurate. In addition, an interfacial heat transfer model is introduced to improve the simulation of the casting process. Advective heat transfer in the liquid is also modelled.

MST/1041     

Adaptive Poly-FEM for the analysis of plane elasticity problems

In this work, we present an adaptive polygonal finite element method (Poly-FEM) for the analysis of two-dimensional plane elasticity problems. The generation of meshes consisting of n ? sided polygonal finite elements is based on the generation of a centroidal Voronoi tessellation (CVT). An unstructured tessellation of a scattered point set, that minimally covers the proximal space around each point in the point set, is generated whereby the method also includes tessellation of nonconvex domains. In this work, we propose a region by region adaptive polygonal element mesh generation. A patch recovery type of stress smoothing technique that utilizes polygonal element patches for obtaining smooth stresses is proposed for obtaining the smoothed finite element stresses. A recovery type a ? posteriori error estimator that estimates the energy norm of the error from the recovered solution is then adopted for the Poly-FEM. The refinement of the polygonal elements is then made on an region by region basis through a refinement index. For the numerical integration of the Galerkin weak form over polygonal finite element domains, we resort to classical Gaussian quadrature applied to triangular subdomains of each polygonal element. Numerical examples of two-dimensional plane elasticity problems are presented to demonstrate the efficiency of the proposed adaptive Poly-FEM.     

Efficient adaptive integration of functions with sharp gradients and cusps in n‐dimensional parallelepipeds

In this paper, we study the efficient numerical integration of functions with sharp gradients and cusps. An adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions. The algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp. The error analysis reveals that for a C⁰ function (derivative discontinuity at a point), a rate of convergence of n + 1 is obtained in . Two applications of the adaptive integration scheme are studied. First, we use the adaptive quadratures for the integration of the regularized Heaviside function—a strongly localized function that is used for modeling sharp gradients. Then the adaptive quadratures are employed in the enriched finite element solution of the all‐electron Coulomb problem in crystalline diamond. The source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei. We show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements. The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in n‐dimensional parallelepiped elements. Copyright © 2012 John Wiley & Sons, Ltd.     

Solution procedures for nonlinear structural dynamic analysis

Abstract

A review of the solution techniques most widely used in nonlinear structural dynamics is presented. For nonlinear transient responses several explicit and implicit direct integration methods are compared with respect to accuracy, stability and computational efficiency. It is concluded that the choice of a suitable method depends upon the nature of the method, the formulation of finite element models, and the problem itself. In general, the Park method seems to be superior to the others in nonlinear dynamic analysis. If equilibrium iterations are performed at each time step, the Newmark — ß= 1/4 method should be preferred.     

Optimal measurement setup for damage detection in piezoelectric plates

An optimization of the excitation-measurement configuration is proposed for the characterization of damage in PZT-4 piezoelectric plates, from a numerical point of view. To perform such an optimization, a numerical method to determine the location and extent of defects in piezoelectric plates is developed by combining the solution of an identification inverse problem, using genetic algorithms and gradient-based methods to minimize a cost functional, and using an optimized finite element code and meshing algorithm. In addition, a semianalytical estimate of the probability of detection is developed and validated, which provides a flexible criterion to optimize the experimental design. The experimental setup is optimized upon several criteria: maximizing the probability of detection against noise effects, ensuring robust search algorithm convergence and increasing the sensitivity to the presence of the defect. The measurement of voltage ? is concluded to provide the highest identifiability, combined with an excitation of the specimen by a mechanical traction transverse to the polarization direction. Sufficient accuracy is predicted for the damage location and sizing under realistic noise levels.     

Autonomous decentralized finite element method and its applications

In this paper, a new solution procedure using the finite element technique in order to solve problems of structure analysis is proposed. This procedure is called the autonomous decentralized finite element method because it is based on the characteristic autonomy and decentralization in life or biological systems (life‐like approach). The fundamental approach is developed according to an idea of cellular automata manipulation by the new neighbourhood model. The finite element method with an algorithm of the relaxation method is adopted as the solution procedure in this approach. The proposed procedure demonstrates that it is a powerful means of numerical analysis for many kinds of structural problems that are structural morphogenesis, structural optimization and structural inverse problems. Our procedure is applied to numerical analysis of three simple plane models: (1) The structural shape analysis problem for the prescribed displacement mode of a truss structure, (2) An adaptive structure remodelling problem on an elastic continuum, (3) An identification problem of thermal conductivity on a continuum. The effectiveness and validity of our idea are shown from their numerical results. Copyright © 2003 John Wiley & Sons, Ltd.     

Isogeometric analysis of shear bands

Numerical modeling of shear bands present several challenges, primarily due to strain softening, strong nonlinear multiphysics coupling, and steep solution gradients with fine solution features. In general it is not known a priori where a shear band will form or propagate, thus adaptive refinement is sometimes necessary to increase the resolution near the band. In this work we explore the use of isogeometric analysis for shear band problems by constructing and testing several combinations of NURBS elements for a mixed finite element shear band formulation. Owing to the higher order continuity of the NURBS basis, fine solution features such as shear bands can be resolved accurately and efficiently without adaptive refinement. The results are compared to a mixed element formulation with linear functions for displacement and temperature and Pian–Sumihara shape functions for stress. We find that an element based on high order NURBS functions for displacement, temperature and stress, combined with gauss point sampling of the plastic strain leads to attractive results in terms of rate of convergence, accuracy and cpu time. This element is implemented with a \(\overline{\hbox {B}}\) -bar strain projection method and is shown to be nearly locking free.     

钢筋混凝土异形柱双向偏心承载力高效算法与程序设计

异形柱结构的发展需要一种高效、精确的承载力求解方法。目前常用的数值积分方法是将截面剖分成二维网格来进行求解,这种方法虽然比较精确,但往往存在计算速度与计算精度之间的矛盾,尤其是计算包含大量构件的结构体系。针对这一问题提出了新的截面数值积分算法。新算法采用特殊的单元剖分技术,实现了有限元法与单元内部解析求解法的有机结合,并且通过构造新的迭代变量加速了迭代运算的过程。与以往算法的对比分析表明,新算法具有更高的运算速度、运算精度以及更为广泛的适用性。     

The post-processing approach in the finite element method—part 1: Calculation of displacements,stresses and other higher derivatives of the displacements

This is the first in a series of three papers in which we discuss a method for ‘post-processing’ a finite element solution to obtain high accuracy approximations for displacements, stresses, stress intensity factors, etc. Rather than take the values of these quantities ‘directly’ from the finite element solution, we evaluate certain weighted averages of the solution over the entire region. These yield approximations are of the same order of accuracy as the strain energy. We obtain error estimates, and also present some numerical examples to illustrate the practical effectiveness of the technique. In the third paper of this series we address the matters of adaptive mesh selection and a posteriori error estimation.     

A combined r-h adaptive strategy based on material forces and error assessment for plane problems and bimaterial interfaces

An r-h adaptive scheme has been proposed and formulated for analysis of bimaterial interface problems using adaptive finite element method. It involves a combination of the configurational force based r-adaption with weighted laplacian smoothing and mesh enrichment by h-refinement. The Configurational driving force is evaluated by considering the weak form of the material force balance for bimaterial inerface problems. These forces assembled at nodes act as an indicator for r-adaption. A weighted laplacian smoothing is performed for smoothing the mesh. The h-adaptive strategy is based on a modifed weighted energy norm of error evaluated using supercovergent estimators. The proposed method applies specific non sliding interface strain compatibility requirements across inter material boundaries consistent with physical principles to obtain modified error estimators. The best sequence of combining r- and h-adaption has been evolved from numerical study. The study confirms that the proposed combined r-h adaption is more efficient than a purely h-adaptive approach and more flexible than a purely r-adaptive approach with better convergence characteristics and helps in obtaining optimal finite element meshes for a specified accuracy.     

膜结构极小曲面找形的一种自适应有限元分析

找形分析是膜结构设计中的关键环节,但在数学上,膜结构的极小曲面找形分析是一个高度非线性问题,一般无法求得其解析解,因此数值方法成为重要工具。近年来,基于单元能量投影法（EEP法）的一维非线性有限元的自适应分析已经取得成功,基于EEP法的二维线性有限元自适应分析也被证实是有效、可靠的。在此基础上,该文提出一种基于EEP法的二维非线性有限元自适应方法,并成功将之应用于膜结构的找形分析。其主要思想是,通过将非线性问题用Newton法线性化,引入现有的二维线性问题的自适应求解技术,进而实现二维有限元自适应分析技术从线性到非线性的跨越,将非线性有限元的自适应分析求解从一维问题拓展到二维问题。该方法兼顾求解的精度和效率,对网格自适应地进行调整,最终得到优化的网格,其解答可按最大模度量逐点满足用户设定的误差限。该文综述介绍了这一进展,并给出数值算例用以表明该方法的可行性和可靠性。     

An h‐hierarchical adaptive procedure for the scaled boundary finite‐element method

The scaled boundary finite‐element method (a novel semi‐analytical method for solving linear partial differential equations) involves the solution of a quadratic eigenproblem, the computational expense of which rises rapidly as the number of degrees of freedom increases. Consequently, it is desirable to use the minimum number of degrees of freedom necessary to achieve the accuracy desired. Stress recovery and error estimation techniques for the method have recently been developed. This paper describes an h‐hierarchical adaptive procedure for the scaled boundary finite‐element method. To allow full advantage to be taken of the ability of the scaled boundary finite‐element method to model stress singularities at the scaling centre, and to avoid discretization of certain adjacent segments of the boundary, a sub‐structuring technique is used. The effectiveness of the procedure is demonstrated through a set of examples. The procedure is compared with a similar h‐hierarchical finite element procedure. Since the error estimators in both cases evaluate the energy norm of the stress error, the computational cost of solutions of similar overall accuracy can be compared directly. The examples include the first reported direct comparison of the computational efficiency of the scaled boundary finite‐element method and the finite element method. The scaled boundary finite‐element method is found to reduce the computational effort considerably. Copyright © 2002 John Wiley & Sons, Ltd.     

Optimal discretizations in adaptive finite element electromagnetics

One of the primary objectives of adaptive finite element analysis research is to determine how to effectively discretize a problem in order to obtain a sufficiently accurate solution efficiently. Therefore, the characterization of optimal finite element solution properties could have significant implications on the development of improved adaptive solver technologies. Ultimately, the analysis of optimally discretized systems, in order to learn about ideal solution characteristics, can lead to the design of better feedback refinement criteria for guiding practical adaptive solvers towards optimal solutions efficiently and reliably. A theoretical framework for the qualitative and numerical study of optimal finite element solutions to differential equations of macroscopic electromagnetics is presented in this study for one-, two- and three-dimensional systems. The formulation is based on variational aspects of optimal discretizations for Helmholtz systems that are closely related to the underlying stationarity principle used in computing finite element solutions to continuum problems. In addition, the theory is adequately general and appropriate for the study of a range of electromagnetics problems including static and time-harmonic phenomena. Moreover, finite element discretizations with arbitrary distributions of element sizes and degrees of approximating functions are assumed, so that the implications of the theory for practical h-, p-, hp- and r-type finite element adaption in multidimensional analyses may be examined. Copyright © 2001 John Wiley & Sons, Ltd.     

Adaptive delamination analysis

A methodology aimed at addressing computational complexity of analyzing delamination in large structural components made of laminated composites is proposed. The classical ply‐by‐ply discretization of individual layers may increase the size of the problem by an order of magnitude in comparison with the laminated shell or plate element meshes. The paper features delamination indicators that pinpoint the onset and propagation of delamination fronts with striking accuracy. Once the location of delamination has been identified, the discrete solution space of the classical laminated plate/shell element is hierarchically enriched by a combination of weak and strong discontinuities to adaptively track the evolution of delamination fronts. The so‐called adaptive s‐method proposed herein is equivalent in terms of approximation space to the extended finite element method but offers sparser matrices and added flexibility in transitioning from weak to strong discontinuities. Numerical examples suggest that despite an overhead that comes with adaptivity, the adaptive s‐method is computationally advantageous over the classical ply‐by‐ply discretization, especially as the problem size increases. Copyright © 2015 John Wiley & Sons, Ltd.