We study the discretization behavior of classical finite element and NURBS approximations on problems of structural vibrations and wave propagation. We find that, on the basis of equal numbers of ...degrees-of-freedom and bandwidth, NURBS have superior approximation properties. In fact, we observe that the high mode behavior of classical finite elements is divergent with the order of approximation, a surprisingly negative result. On the other hand, NURBS offer almost spectral approximation properties, and all modes converge with increasing order of approximation.
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
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-continuous “
p-method,” whereas a judicious insertion of
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-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
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-continuous finite elements and therefore further studies are warranted.
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.
In 2015–2016, record temperatures triggered a pan-tropical episode of coral bleaching. In the southern hemisphere summer of March–April 2016, we used aerial surveys to measure the level of bleaching ...on 1,156 individual reefs throughout the 2,300 km length of the Great Barrier Reef, the world’s largest coral reef system. The accuracy of the aerial scores was ground-truthed with detailed underwater surveys of bleaching at 260 sites (104 reefs), allowing us to compare aerial and underwater bleaching data with satellite-derived temperatures and with associated model predictions of bleaching. The severity of bleaching on individual reefs in 2016 was tightly correlated with the level of local heat exposure: the southernmost region of the Great Barrier Reef escaped with only minor bleaching because summer temperatures there were close to average. Gradients in nutrients and turbidity from inshore to offshore across the Great Barrier Reef had minimal effect on the severity of bleaching. Similarly, bleaching was equally severe on reefs that are open or closed to fishing, once the level of satellite-derived heat exposure was accounted for. The level of post-bleaching mortality, measured underwater after 7–8 months, was tightly correlated with the aerial scores measured at the peak of bleaching. Similarly, reefs with a high aerial bleaching score also experienced major shifts in species composition due to extensive mortality of heat-sensitive species. Reefs with low bleaching scores did not change in composition, and some showed minor increases in coral cover. Two earlier mass bleaching events occurred on the Great Barrier Reef in 1998 and 2002, that were less severe than 2016. In 2016, <9% of scored reefs had no bleaching, compared to 42% in 2002 and 44% in 1998. Conversely, the proportion of reefs that were severely bleached (>60% of corals affected) was four times higher in 2016. The geographic footprint of each of the three events is distinctive, and matches satellite-derived sea surface temperature patterns. Our aerial surveys indicate that past exposure to bleaching in 1998 and 2002 did not lessen the severity of bleaching in 2016. This data set of aerial bleaching scores provides a historical baseline for comparison with future bleaching events. No copyright restrictions apply to the use of this data set other than citing this publication.
We study NURBS-based isogeometric analysis of contact problems and compare with standard
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-continuous Lagrange finite elements. A knot-to-surface (KTS) algorithm is developed to treat the contact ...constraints with NURBS contact surface discretizations. Qualitative studies deliver satisfactory results for various finite deformation frictionless thermoelastic contact problems. Quantitative studies based on the Hertz problem suggest the need for a relaxation of the mechanical contact constraints that appear in the standard KTS approach. The improved mortar-based KTS algorithm delivers robust and accurate results for NURBS discretizations. Based on numerical examples, we conclude that NURBS-based isogeometric analysis is a viable technology for contact problems and offers potential accuracy as well as convergence improvements over
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-continuous finite elements.
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.
Tropical corals live close to their upper thermal limit making them vulnerable to unusually warm summer sea temperatures. The resulting thermal stress can lead to breakdown of the coral-algal ...symbiosis, essential for the functioning of reefs, and cause coral bleaching. Mass coral bleaching is a modern phenomenon associated with increases in reef temperatures due to recent global warming. Widespread bleaching has typically occurred during El Niño events. We examine the historical level of stress for 100 coral reef locations with robust bleaching histories. The level of thermal stress (based on a degree heating month index, DHMI) at these locations during the 2015-2016 El Niño was unprecedented over the period 1871-2017 and exceeded that of the strong 1997-1998 El Niño. The DHMI was also 5 times the level of thermal stress associated with the 'pre-industrial', 1877-1878, El Niño. Coral reefs have, therefore, already shown their vulnerability to the modest (~0.92 °C) global warming that has occurred to date. Estimates of future levels of thermal stress suggest that even the optimistic 1.5 °C Paris Agreement target is insufficient to prevent more frequent mass bleaching events for the world's reefs. Effectively, reefs of the future will not be the same as those of the past.
We develop a local refinement algorithm for analysis-suitable T-splines which does not produce excessive propagation of control points. We then demonstrate its use as an adaptive framework for ...isogeometric analysis. Analysis-suitable T-splines are a class of T-splines which are linearly independent and form a partition of unity. These properties, coupled with local refinement, make this class of T-splines appealing as a basis for isogeometric analysis.
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.
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