Isogeometric Collocation: Cost Comparison with Galerkin Methods and Extension to Adaptive Hierarchical NURBS Discretizations

Collocation is based on the discretization of the strong form of the underlying partial differential equations, which requires basis functions of sufficient order and smoothness. Consequently, the use of isogeometric analysis (IGA) for collocation suggests itself, since splines can be readily adjusted to any order in polynomial degree and continuity required by the differential operators. In addition, they can be generated for domains of arbitrary geometric and topological complexity, directly linked to and fully supported by CAD technology. The major advantage of isogeometric collocation over Galerkin type IGA is the minimization of the computational effort for numerical quadrature. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Static and dynamic analyses of isogeometric curvilinearly stiffened plates

The isogeometric analysis (IGA) is a new approach which builds a seamless connection between Computer Aided Design (CAD) and Computer Aided Engineering (CAE). This approach which uses the B-Splines or the Non-Uniform Rational B-Splines (NURBS) as a geometric representation of the object is a discretization technology for numerical analysis. The IGA has advantages of capturing exact geometry and making the flexibility of refinement, which results in higher calculation accuracy. To study the static and dynamic characteristics of curvilinearly stiffened plates, the NURBS based isogeometric analysis approach is developed in this paper. We use this approach to analyze the static deformation, the free vibration and the vibration behavior in the presence of in-plane loads of curvilinearly stiffened plates. Furthermore, the large deformation and the large amplitude vibration of the curvilinearly stiffened plates are also studied based on the von Karman's large deformation theory. One of the superiorities of the present method in the analysis of the stiffened plates is that the element number is much less than commercial finite element software, whereas another advantage is that the mesh refinement process is much more convenient compared with traditional finite element method (FEM). Some numerical examples are shown to validate the correctness and superiority of the present method by comparing with the results from commercial software and finite element analysis.     

On the formulation of a BEM in the Bézier–Bernstein space for the solution of Helmholtz equation

This paper proposes a novel boundary element approach formulated on the Bézier-Bernstein basis to yield a geometry-independent field approximation. The proposed method is geometrically based on both computer aid design (CAD) and isogeometric analysis (IGA), but field variables are independently approximated from the geometry. This approach allows the appropriate approximation functions for the geometry and variable field to be chosen. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The solution of the element interpolation problem in the Bézier–Bernstein space defines generalised Lagrange interpolation functions that are used as element shape functions. The resulting Bernstein–Vandermonde matrix related to the Bézier–Bernstein interpolation problem is inverted using the Newton-Bernstein algorithm. The applicability of the proposed method is demonstrated solving the Helmholtz equation over an unbounded region in a two-and-a-half dimensional (2.5D) domain.     

Shape Gradient for Isogeometric Structural Design

The transfer of geometrical data from CAD (Computer Aided Design) to FEA (Finite-Element Analysis) is a bottleneck of automated design optimization procedures, yielding a loss of accuracy and cumbersome software couplings. Isogeometric analysis methods propose a new paradigm that allows one to overcome these difficulties by using a unique geometrical representation, yielding a direct relationship between geometry and analysis. In this study, in the framework of linear elasticity problems, we investigate its use for sensitivity analysis and, more specifically, shape gradient computations.     

A Neuro-Fuzzy Approach to Optimize Hierarchical Recurrent Fuzzy Systems

To simplify the definition of fuzzy systems or to reduce its complexity hierarchical structures can be used. Thus, more transparent rule bases that are also easier to maintain can be designed. Furthermore, it is sometimes necessary to use time delayed input or to reuse time delayed output from the fuzzy system itself to obtain a rule base that describes the analyzed problem appropriately. This leads to hierarchical recurrent architectures that have increased approximation capabilities since they are able to store information of the past. In this article we present a neuro-fuzzy model that can be used to optimize hierarchical recurrent fuzzy rule bases if training data is available. Furthermore, we present an approach to learn initial rule bases from data using rule templates.     

高维有限元空间中的谱基

高维有限元空间中的谱基施锡泉（大连理工大学数学科学研究所）ＡＣＯＮＳＴＲＵＣＴＩＶＥＭＥＴＨＯＤＯＦＨＩＧＨ－ＤＩＭＥＮＳＩＯＮＡＬＨＩＥＲＡＲＣＨＩＣＡＬＢＡＳＩＳ￥ＳｈｉＸｉ－ｑｕａｎ（Ｉｎｓｔ．ｏｆＭａｔｈ．，ＤａｌｉａｎＵｎｉｖｅｒｓｉｔｙｏ...     

Material optimization of tri-directional functionally graded plates by using deep neural network and isogeometric multimesh design approach

The paper is aimed at enhancing computational performance for optimizing the material distribution of tri-directional functionally graded (FG) plates. We exploit advantages of using a non-uniform rational B-spline (NURBS) basis function for describing material distribution varying through all three directions of functionally graded (FG) plates. Two-dimensional free vibration and buckling behaviors of multi-directional (1D, 2D and 3D) FG plates analyzed by using a combination of generalized shear deformation theory (GSDT) and isogeometric analysis (IGA) is first proposed. This approach can help to save a significant amount of computational cost while still ensure the accuracy of the solutions. The effectiveness and reliability of the present method are demonstrated by comparing it to other methods in the literature. The obtained results are in excellent agreement with the reference ones. More importantly, data sets consisting of input-output pairs are randomly generated from the analysis process through iterations for the training process in deep neural networks (DNN). DNN is utilized as an analysis tool to supplant finite element analysis to reduce computational cost. By using DNN, behaviors of the multi-directional FG plates are directly predicted from those material distributions. Optimal material distributions of tri-directional FG plates under free vibration or compression in various volume fraction constraints are found by using modified symbiotic organisms search (mSOS) algorithm for the first time. Moreover, an isogeometric multimesh design technique is also used to diminish a large number of design variables in optimization. Optimal results obtained by DNN are compared with those of IGA to verify the effectiveness of the proposed method.     

Isogeometric vibration and buckling analyses of curvilinearly stiffened composite laminates

Isogeometric analysis (IGA) with the polynomial splines over hierarchical T-meshes (PHT-splines) is used to provide an efficient tool capable of carrying out the vibration and buckling analyses of the stiffened laminates. IGA offers increased accuracy and efficiency using the PHT-splines, which represent exact geometry of the stiffeners and make the refinement more flexible near the areas where the stiffeners and composite plate are connected. Numerical examples are given to validate the correctness and superiority of the present method, comparing with the results from existing literatures and commercial softwares. Besides, the influences of the orientation, curvature, location and cross-section size of the stiffeners on the natural frequencies and buckling loads are also studied. The results show that the optimization of the shape and size of the stiffeners has an important effect on the vibration and buckling characteristics of stiffened laminates.     

Minimal invariant Markov basis for sampling contingency tables with fixed marginals

In this paper we define an invariant Markov basis for a connected Markov chain over the set of contingency tables with fixed marginals and derive some characterizations of minimality of the invariant basis. We also give a necessary and sufficient condition for uniqueness of minimal invariant Markov bases. By considering the invariance, Markov bases can be presented very concisely. As an example, we present minimal invariant Markov bases for all 2 × 2 × 2 × 2 hierarchical models. The invariance here refers to permutation of indices of each axis of the contingency tables. If the categories of each axis do not have any order relations among them, it is natural to consider the action of the symmetric group on each axis of the contingency table. A general algebraic algorithm for obtaining a Markov basis was given by Diaconis and Sturmfels (The Annals of Statistics, 26, 363–397, 1998). Their algorithm is based on computing Gröbner basis of a well-specified polynomial ideal. However, the reduced Gröbner basis depends on the particular term order and is not symmetric. Therefore, it is of interest to consider the properties of invariant Markov basis.     

A reduced beam and joint concept modeling approach to optimize global vehicle body dynamics

Ideally, NVH simulations become available already in the concept phase of vehicle development. The initial computer-aided design (CAD) can then be improved (by already including countermeasures), and the feasibility to balance NVH with other performance attributes is increased. In this early design stage, when exact geometrical information is not or scarcely available, conventional virtual prototyping techniques based on detailed CAD and FE models are not directly applicable. A state-of-the-art overview of concept NVH simulation methods in vehicle industry is given.This paper then presents a “Reduced Beam and Joint Modeling” approach to analyze and optimize the global bending and torsion modes of a vehicle body. Concept modifications in the body beam-like sections and in the joints are analyzed using the body reduced modal model. This small-sized model can be used to quickly and accurately optimize the low-frequency vehicle performance. The modifications are considered with respect to the existing (predecessor) model. Equivalent beam properties are estimated from the body FE model; modifications in the beam-like sections are then implemented with beam elements from a standard FE library. The joint modifications are considered through static superelements: stiffness formulations between the end points of the joint connected to the beam layout. The validity of the approach is first demonstrated on simple example models. An industrial vehicle BIW application case is subsequently presented. A beam and joint layout is created, and used for a fast sensitivity analysis to identify suitable modifications to improve the global modes. Next, two application cases are presented. First, a fast optimization analysis is performed to optimize the global body modes. Subsequently, suitable physical modifications are identified and applied to the full FE model; it is shown that the effect of these physical modifications is accurately predicted with the fast sensitivity analysis.     

A study of isogeometric analysis for scalar convection–diffusion equations

Isogeometric analysis (IGA), in combination with the streamline upwind Petrov–Galerkin (SUPG) stabilization, is studied for the discretization of steady-state convection–diffusion equations. Numerical results obtained for the Hemker problem are compared with results computed with the SUPG finite element method of the same order. Using an appropriate parameterization for IGA, the computed solutions are much more accurate than those obtained with the finite element method, both in terms of the size of spurious oscillations and of the sharpness of layers.     

线性规划中的退化最优基图

Abstract. The basis graph G for a linear programming consists of all bases under pivot transfor-mations. A degenerate optimal basis graph G is a subgraph of G induced by all optimal bases ata degenerate optimal vertex x. In this paper, several conditions for the characterization of G“are presented.     

Phase-field approach to fracture for finite-deformation contact problems

The present contribution deals with a variationally consistent Mortar contact algorithm applied to a phase-field fracture approach for finite deformations, see [4]. A phase-field approach to fracture allows for the numerical simulation of complex fracture patterns for three dimensional problems, extended recently to finite deformations (see [2] for more details). In a nutshell, the phase-field approach relies on a regularization of the sharp (fracture-) interface. In order to improve the accuracy, a fourth-order Cahn-Hilliard phase-field equation is considered, requiring global C¹ continuity (see [1]), which will be dealt with using an isogeometrical analysis (IGA) framework. Additionally, a newly developed hierarchical refinement scheme is applied to resolve for local physical phenomena e.g. the contact zone (see [3] for more details). The Mortar method is a modern and very accurate numerical method to implement contact boundaries. This approach can be extended in a straightforward manner to transient phase-field fracture problems. The performance of the proposed methods will be examined in a representative numerical example. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An adaptive wavelet space‐time SUPG method for hyperbolic conservation laws

This article concerns with incorporating wavelet bases into existing streamline upwind Petrov‐Galerkin (SUPG) methods for the numerical solution of nonlinear hyperbolic conservation laws which are known to develop shock solutions. Here, we utilize an SUPG formulation using continuous Galerkin in space and discontinuous Galerkin in time. The main motivation for such a combination is that these methods have good stability properties thanks to adding diffusion in the direction of streamlines. But they are more expensive than explicit semidiscrete methods as they have to use space‐time formulations. Using wavelet bases we maintain the stability properties of SUPG methods while we reduce the cost of these methods significantly through natural adaptivity of wavelet expansions. In addition, wavelet bases have a hierarchical structure. We use this property to numerically investigate the hierarchical addition of an artificial diffusion for further stabilization in spirit of spectral diffusion. Furthermore, we add the hierarchical diffusion only in the vicinity of discontinuities using the feature of wavelet bases in detection of location of discontinuities. Also, we again use the last feature of the wavelet bases to perform a postprocessing using a denosing technique based on a minimization formulation to reduce Gibbs oscillations near discontinuities while keeping other regions intact. Finally, we show the performance of the proposed combination through some numerical examples including Burgers’, transport, and wave equations as well as systems of shallow water equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2062–2089, 2017     

On the global convergence of an SLP–filter algorithm that takes EQP steps

 A global convergence proof is presented for a class of trust region filter–type methods for nonlinear programming. Such methods are characterized by their use of the dominance concept of multiobjective optimization, instead of a penalty parameter whose adjustment can be problematic. The methods are based on successively solving linear programming subproblems for which effective software is readily available. The methods also permit the use of steps calculated on the basis of an equality constrained quadratic programming model, which enables rapid convergence to take place for problems in which second order information is important. The proof technique is presented in a fairly general context, allowing a range of specific algorithm choices associated with choosing the quadratic model, updating the trust region radius and with feasibility restoration. Received: July 4, 2001 / Accepted: December 19, 2002 Published online: March 21, 2003 Mathematics Subject Classification (2000): 20E28, 20G40, 20C20     

Discrete Stochastic Arithmetic for Validating Results of Numerical Software

The Discrete Stochastic Arithmetic DSA is a probabilistic approach for round-off error propagation. After a brief review of the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method, which is the basis of DSA, the concept of the “informatical zero”, also called “computational zero”, is defined. The stochastic order relations of the DSA are presented. The DSA is the joint use of the synchronous implementation of the CESTAC method and the stochastic order relations. After having summarized the asynchronous implementation of the CESTAC method, which has been used in the Prosolver software, and which has been legitimately criticized, the synchronous implementation is presented. Then the CADNA (Control of Accuracy and Debugging for Numerical Application) library which implements the DSA arithmetic is presented. It is shown that this library is able to dynamically control the validity of the hypotheses which must hold so that results provided by CESTAC method are reliable. If the hypotheses do not hold then warnings are printed in a special file. The user is informed that numerical anomalies have been detected. Depending on these warnings the user may conclude either that the results obtained are not reliable and that they cannot be correctly computed with this computer, or he may try to debug his code. It is shown that the numerical examples that support the criticisms and which make the Prosolver software fail, do not jeopardize the CADNA library.     

一类含幅、相等式的非线性方程组的线性化及其应用

一类有理分式复函数G(jω)=1+a₁jω+a₂(jω)2+…+a_m(jω)n/b₀+b₁jω+b₂jω+…+b_n(jω)n常被用来描述系统的性能.当其有关的幅值(模)和相角(幅角)的数据能获取时,可以对G(jω)进行综合.诸未知系数a_i及b_i(i=0,1,…,m,…,n)将通过解一类含幅、相等式的非线性代数方程组_i i=1,2,…,r.确解的初值[9].本文通过简单的数学处理,将这类非线性方程组完全线性化为同维的线性方程组,从而得以直接解.此法还可以推广到分母含纯(jω)因子和e-jωi0因子的系统.文章最后通过例子简介了这种方法在控制工程领域的应用.     

On the multi-level splitting of finite element spaces

Summary In this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases. We show that the condition number of such a stiffness matrix behaves like O((log )²) where is the condition number of the stiffness matrix with respect to a nodal basis. In the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like instead of for a nodal basis. The proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization. Especially we do not need the quasiuniformity of the employed triangulations. As the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis, our results mean that the method of conjugate gradients needs onlyO(log n) steps andO(n log n) computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures. Heren denotes the dimension of the finite element space and of the discrete linear problem to be solved.     

Analysis and Application of the Rational B-Spline Finite Element Method in 2D

Non-Uniform Rational B-Splines (NURBS) are basis functions used in CAD software to describe exact geometric models. The implementation of these basis functions in the context of the Finite Element Analysis (FEA) is known as isogeometric analysis. The concept and definition of NURBS is briefly presented here. Since these functions are implemented as shape functions for the isogeometric analysis, the refinement strategies are discussed. The example of an infinite plate with circular hole serves as a benchmark. Finally, isogeometric analysis is applied to gradient elasticity since NURBS functions are of higher continuity and this is required in gradient elasticity. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bivariate hierarchical Hermite spline quasi-interpolation

Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low cost and accurate approximations of a given function. In order to provide an efficient adaptive approximation scheme in the bivariate setting, we consider quasi-interpolation in hierarchical spline spaces. In particular, we study and experiment the features of the hierarchical extension of the tensor-product formulation of the Hermite BS quasi-interpolation scheme. The convergence properties of this hierarchical operator, suitably defined in terms of truncated hierarchical B-spline bases, are analyzed. A selection of numerical examples is presented to compare the performances of the hierarchical and tensor-product versions of the scheme.