Abstract The concept of isogeometric analysis is proposed. Basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model. For purposes of analysis, the basis is refined and/or its order elevated without changing the geometry or its parameterization. Analogues of finite element h - and p -refinement schemes are presented and a new, more efficient, higher-order concept, k -refinement, is introduced. Refinements are easily implemented and exact geometry is maintained at all levels without the necessity of subsequent communication with a CAD (Computer Aided Design) description. In the context of structural mechanics, it is established that the basis functions are complete with respect to affine transformations, meaning that all rigid body motions and constant strain states are exactly represented. Standard patch tests are likewise satisfied. Numerical examples exhibit optimal rates of convergence for linear elasticity problems and convergence to thin elastic shell solutions. A k -refinement strategy is shown to converge toward monotone solutions for advection–diffusion processes with sharp internal and boundary layers, a very surprising result. It is argued that isogeometric analysis is a viable alternative to standard, polynomial-based, finite element analysis and possesses several advantages.

Abstract : This article presents an introduction to multiscale and stabilized methods, which represent unified approaches to modeling and numerical solution of fluid dynamic phenomena. Finite element applications are emphasized but the ideas are general and apply to other numerical methods as well. (They have been used in the development of finite difference, finite volume, and spectral methods, in addition to finite element methods.) The analytical ideas are first illustrated for time-harmonic ...

Last. Wolfgang A. Wall(TUM: Technische Universität München)H-Index: 62

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The analysis of large-scale nonlinear shell problems asks for parallel simulation approaches. One crucial part of effi- cient and well scalable parallel FE-simulations is the solver for the system of equations. Due to the inherent suitability for parallelization one is verymuch directed towards preconditioned iterative solvers. However thin-walled-structures discretized byfinite elements lead to ill-conditioned system matrices and therefore performance of iterative solvers is generallypoor. This...

#1Ernst Rank(TUM: Technische Universität München)H-Index: 60

#2Alexander Düster(TUM: Technische Universität München)H-Index: 31

Last. O.T. Bruhns(RUB: Ruhr University Bochum)H-Index: 1

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In this article the p-version finite element method is applied to thin-walled structures. Two different hierarchic element formulations are compared, a shell approach as well as a shell-like, solid formulation. Both approaches are compared for linear elastic and elastoplastic problems. Special emphasis is placed on the efficiency as well as on determining the area of application for both formulations.

Last. Ekkehard Ramm(University of Stuttgart)H-Index: 53

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The present study provides an overview of modeling and discretization aspects in finite element analysis of thin-walled structures. Shell formulations based upon derivation from three-dimensional continuum mechanics, the direct approach, and the degenerated solid concept are compared, highlighting conditions for their equivalence. Rather than individually describing the innumerable contributions to theories and finite element formulations for plates and shells, the essential decisions in modelin...

This paper describes modern techniques used to solve contact problems within Computational Mechanics. On the basis of a continuum description of contact, the mathematical structure of the contact formulation for finite element methods is derived. Emphasis is also placed on the constitutive behavior at the contact interface for normal and tangential (frictional) contact. Furthermore, different discretization schemes currently applied to solve engineering problems are formulated for small and fini...

#1Barna A. Szabó(WashU: Washington University in St. Louis)H-Index: 27

#2Alexander Düster(TUM: Technische Universität München)H-Index: 31

Last. Ernst Rank(TUM: Technische Universität München)H-Index: 60

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In the p-version of the finite element method, the triangulation is fixed and the degree p, of the piecewise polynomial approximation, is progressively increased until some desired level of precision is reached. In this paper, we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation. These properties lead to an a priori estimate of the asymptotic rate of convergence of the p-version. The estimate shows that the p-vers...

This paper presents a multiscale method that yields a stabilized finite element formulation for the advection–diffusion equation. The multiscale method arises from a decomposition of the scalar field into coarse (resolved) scale and fine (unresolved) scale. The resulting stabilized formulation possesses superior properties like that of the SUPG and the GLS methods. A significant feature of the present method is that the definition of the stabilization term appears naturally, and therefore the fo...

Abstract We investigate, for linear and higher-order elements, various ways of calculating the advective limit of the stabilization parameter (“ τ ”) used in the streamline-upwind/Petrov–Galerkin (SUPG) formulation of flow problems. In the context of a pure advection test problem, we compare the “UGN-based”, element-matrix-based, and element-node-based calculations of the advective limit of the τ . Our investigation shows that the performances of the “UGN-based” and element-matrix-based τ defini...

Abstract The enhanced-discretization space–time technique (EDSTT) was developed for the purpose of being able to, in the context of a space–time formulation, enhance the time-discretization in regions of the fluid domain requiring smaller time steps. Such requirements are often encountered in time-accurate computations of fluid–structure interactions, where the time-step size required by the structural dynamics part is smaller, and carrying out the entire computation with that time-step size wou...

#2Weiyin Ma(CityU: City University of Hong Kong)H-Index: 22

Abstract null null UE-splines are generalizations of uniform polynomial splines, trigonometric splines, hyperbolic splines and exponential splines defined as parametric splines in non-rational form. At the same time, they can exactly represent a wide class of basic analytic shapes commonly used in engineering applications. Further, UE-splines with uniform knot intervals can be conveniently generated by subdivision methods. In this paper, we use them to model many kinds of basic analytic shapes a...

Last. Shaowei Yang(DUT: Dalian University of Technology)

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Abstract null null null null null This paper presents a numerical model based on the Isogeometric Analysis (IGA), null Boundary Element Method null (BEM) and the second order Runge–Kutta (RK2) method for solving the nonlinear sloshing problem in a two-dimensional (2D) tank. In this study, the IGABEM is employed to solve the null null Laplace equation null for the null null velocity potentials null of the liquid, and the null null singular integral null null is calculated by the Radial Integral M...

Abstract null null null null A key advantage of isogeometric null discretizations null null is their accurate and well-behaved null eigenfrequencies null and null null null eigenmodes . For degree two and higher, however, a few spurious modes appear that possess inaccurate frequencies, denoted as “outliers”. The outlier frequencies and corresponding modes are at the root of several efficiency and robustness issues in isogeometric analysis. One example is explicit dynamics where outlier frequenci...

Space separation within the Proper Generalized Decomposition—PGD—rationale allows solving high dimensional problems as a sequence of lower dimensional ones. In our former works, different geometrical transformations were proposed for addressing complex shapes and spatially non-separable domains. Efficient implementation of separated representations needs expressing the domain as a product of characteristic functions involving the different space coordinates. In the case of complex shapes, more s...

Last. Jörg Peters(UF: University of Florida)H-Index: 29

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Abstract null null When the full-scale storing and retrieving of volumetric models is cost prohibitive, intersection queries require intelligent access to pieces generated on demand. To conform to a given curved outer shape without clipping, such models are often the result of a non-linear free-form deformation applied to a geometrically simpler, canonical model. The additional challenge is then to relate the intersection query back to the pieces of the pre-image of the conforming curved model. ...