We develop new quadrature rules for isogeometric analysis based on the solution of a local nonlinear problem. A simple and robust algorithm is developed to determine the rules which are exact for ...important B-spline spaces of uniform and geometrically stretched knot spacings. We consider both periodic and open knot vector configurations and illustrate the efficiency of the rules on selected boundary value problems. We find that the rules are almost optimally efficient, but much easier to obtain than optimal rules, which require the solution of global nonlinear problems that are often ill-posed.
We initiate the study of efficient quadrature rules for NURBS-based isogeometric analysis. A rule of thumb emerges, the “half-point rule”, indicating that optimal rules involve a number of points ...roughly equal to half the number of degrees-of-freedom, or equivalently half the number of basis functions of the space under consideration. The half-point rule is independent of the polynomial order of the basis. Efficient rules require taking into account the precise smoothness of basis functions across element boundaries. Several rules of practical interest are obtained, and a numerical procedure for determining efficient rules is presented.
We compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics. The new rules represent improvements over those used previously in isogeometric analysis.
GeoPDEs (
http://geopdes.sourceforge.net) is a suite of free software tools for applications on Isogeometric Analysis (IGA). Its main focus is on providing a common framework for the implementation ...of the many IGA methods for the discretization of partial differential equations currently studied, mainly based on B-Splines and Non-Uniform Rational B-Splines (NURBS), while being flexible enough to allow users to implement new and more general methods with a relatively small effort. This paper presents the philosophy at the basis of the design of
GeoPDEs and its relation to a quite comprehensive, abstract definition of IGA.
We investigate the effects of smoothness of basis functions on solution accuracy within the isogeometric analysis framework. We consider two simple one-dimensional structural eigenvalue problems and ...two static shell boundary value problems modeled with trivariate NURBS solids. We also develop a local refinement strategy that we utilize in one of the shell analyses. We find that increased smoothness, that is, the “
k-method,” leads to a significant increase in accuracy for the problems of structural vibrations over the classical
C
0
-continuous “
p-method,” whereas a judicious insertion of
C
0
-continuous surfaces about singularities in a mesh otherwise generated by the
k-method, usually outperforms a mesh in which all basis functions attain their maximum level of smoothness. We conclude that the potential for the
k-method is high, but smoothness is an issue that is not well understood due to the historical dominance of
C
0
-continuous finite elements and therefore further studies are warranted.
Isogeometric analysis of structural vibrations Cottrell, J.A.; Reali, A.; Bazilevs, Y. ...
Computer methods in applied mechanics and engineering,
08/2006, Letnik:
195, Številka:
41
Journal Article
Recenzirano
Odprti dostop
This paper begins with personal recollections of John H. Argyris. The geometrical spirit embodied in Argyris’s work is revived in the sequel in applying the newly developed concept of isogeometric ...analysis to structural vibration problems. After reviewing some fundamentals of isogeometric analysis, application is made to several structural models, including rods, thin beams, membranes, and thin plates. Rotationless beam and plate models are utilized as well as three-dimensional solid models. The concept of
k-refinement is explored and shown to produce more accurate and robust results than corresponding finite elements. Through the use of nonlinear parameterization, “optical” branches of frequency spectra are eliminated for
k-refined meshes. Optical branches have been identified as contributors to Gibbs phenomena in wave propagation problems and the cause of rapid degradation of higher modes in
p-method finite elements. A geometrically exact model of the NASA Aluminum Testbed Cylinder is constructed and frequencies and mode shapes are computed and shown to compare favorably with experimental results.
► We continue the development of Collocation methods for IGA. ► We focus on linear elasticity problems in any dimension, and develop a variational approach to IGA collocation. ► We treat NURBS ...multi-patch configurations, including various boundary and interface conditions. ► We investigate time dependent problems and explicit dynamic analysis.
We extend the development of collocation methods within the framework of Isogeometric Analysis (IGA) to multi-patch NURBS configurations, various boundary and patch interface conditions, and explicit dynamic analysis. The methods developed are higher-order accurate, stable with no hourglass modes, and efficient in that they require a minimum number of quadrature evaluations. The combination of these attributes has not been obtained previously within standard finite element analysis.
We present an LES-type variational multiscale theory of turbulence. Our approach derives completely from the incompressible Navier–Stokes equations and does not employ any
ad hoc devices, such as ...eddy viscosities. We tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows. In the calculations, we employed linear, quadratic and cubic NURBS. A dispersion analysis of simple model problems revealed NURBS elements to be superior to classical finite elements in approximating advective and diffusive processes, which play a significant role in turbulence computations. The numerical results are very good and confirm the viability of the theoretical framework.
We herein use a (vectorial) phase-field model description of the evolution of ferroelectric domains to be coupled with the equations of electroelasticity, with the aim of building a simulation ...framework for electromechanically active materials. The governing equations of the coupled model are discretized in strong form by means of isogeometric collocation and numerically solved by a staggered explicit approach. This is the first time that isogeometric collocation is used for such a complex problem, comprising a vectorial phase-field model for polarization coupled with the equations describing the electrical and the mechanical response of the system. Several numerical experiments are carried out to test the behavior of the adopted simulation framework. The obtained results are excellent and propose isogeometric collocation as an inexpensive but very accurate alternative to standard finite element discretizations also for complex coupled problems.
The objective of the present work is to develop efficient, higher-order space- and time-accurate, methods for structural dynamics. To this end, we present a family of explicit isogeometric ...collocation methods for structural dynamics that are obtained from predictor–multicorrector schemes. These methods are very similar in structure to explicit finite-difference time-domain methods, and in particular, they exhibit similar levels of computational cost, ease of implementation, and ease of parallelization. However, unlike finite difference methods, they are easily extended to non-trivial geometries of engineering interest. To examine the spectral properties of the explicit isogeometric collocation methods, we first provide a semi-discrete interpretation of the classical predictor–multicorrector method. This allows us to characterize the spatial and modal accuracy of the isogeometric collocation predictor–multicorrector method, irrespective of the considered time-integration scheme, as well as the critical time step size for a particular explicit time-integration scheme. For pure Dirichlet problems, we demonstrate that it is possible to obtain a second-order-in-space scheme with one corrector pass, a fourth-order-in-space scheme with two corrector passes, and a fifth-order-in-space scheme with three corrector passes. For pure Neumann and mixed Dirichlet–Neumann problems, we demonstrate that it is possible to obtain a second-order-in-space scheme with one corrector pass and a third-order-in-space scheme with two corrector passes, and we observe that fourth-order-in-space accuracy may be obtained pre-asymptotically with three corrector passes. We then present second-order-in-time, fourth-order-in-time, and fifth-order-in-time fully discrete predictor–multicorrector algorithms that result from the application of explicit Runge–Kutta methods to the semi-discrete isogeometric collocation predictor–multicorrector method. We confirm the accuracy of the family of explicit isogeometric collocation methods using a suite of numerical examples.
•Explicit predictor–multicorrector isogeometric collocation methods are presented.•A semi-discrete predictor–multicorrector reinterpretation enables spectral analysis.•High-order accuracy in space is observed using a small number of corrector passes.•Higher-order accuracy in time is achieved using explicit Runge–Kutta methods.•Numerical examples confirm the accuracy of the family of methods.
PDF