Isogeometric analysis (IGA) has been recently introduced as a viable alternative to the standard, polynomial-based finite element analysis. One of the fundamental performance issues of the isogeometric analysis is the quadrature of individual components of the discretized governing differential equation. The capability of the isogeometric analysis to easily adopt basis functions implies that high order numerical quadrature schemes must be employed. This may become computationally prohibitive because the evaluation of the high degree basis functions and/or their derivatives at individual integration points is quite demanding. The situation tends to be critical in three-dimensional space where the total number of integration points can increase dramatically. The aim of this paper is to compare computational efficiency of several numerical quadrature concepts which are nowadays available in the isogeometric analysis. Their performance is asessed on the assembly of stifness matrix of B-spline based problems with special geometrical arrangement allowing to determine minimum number of integration points leading to exact results. and Článek zahrnuje seznam literatury a na str. 257-259 Appendix
Modification of the Finite Element Methode (FEM) based on the different types spline shape function is an up-to-date strategy for numerical solution of partial differential equations (PDEs). This approach has an advantage that the geometry in Computer Graphics framework and approximation of fields of unknown quantities in FEM are described by the same technique. The spline variant of FEM is often called the Isogeometric Analysis (IGA). Another benefit of this numerical solution of PDEs is that the approximation of unknown quantities is smooth. It is an outcome of higher-order continuity of spline basis functions. It was shown, that IGA produces outstanding convergence rate and also appropriate frequency errors. Polynomial spline (Cp-1 continuous piecewise polynomials, p ≥ 2) shape functions produce low dispersion errors and moreover, dispersion spectrum of unbounded domains does not include optical modes unlike FEM based on the higher-order C0 continuous Lagrange interpolation polynomials.In this contribution, the B-spline (NURBS with uniform weights) shape functions in the FEM framework are tested in the numerical solution of free vibration of an elastic block. The main attention is paid to the comparison of convergence rate and accuracy of IGA with the classical Lagrangian FEM, Ritz method and experimental data. and Obsahuje seznam literatury
Isogeometric analysis is a quickly emerging alternative ot the standard, polynomial-based finite element analysis. It is only the question of time, when it will be implemented into major software packages and will be intensively used by engineering community to the analysis of complex realistic problems. Computational demands of such analyses, that may likely exceed the capacity of a single computerk can be parallel processing requires usuall an appropriate decomposition of the investigated problem to the individual processing units. In the case of he isogeometric analysis, the decomposition corresponds to the spatial partitioning of the underlying spatial discretization. While there are several matured graphs-based decomposers which can be readily applied to the subdivison of finite element meshes, their use in the context of the isogeometric analysis is not straightforward because of a rather complicated construction of the graph corresponding to the computational isogeometric mesh. In this paper, a new technology for the construction of the dual graph of a two-dimensional NURBS-based (non-uniform rational B-spline) isogeometric mesh is introduced. This makes the partitioning of the isogeometric meshes for parallel processing accessible for the standard graph-based partitioning of the isogeometric meshes for parallel processing accessible for the standard graph-based partitioning approaches. and Obsahuje seznam literatury