Belytschko T. - Introduction (779635), страница 88
Текст из файла (страница 88)
Constant-strain triangular and tetrahedralelements are characterized by severe volumetric locking in two-dimensional plane strainproblems and in three dimensions. They also manifest stiff behavior in many othercases, such as beam bending. For arbitrary arrangements of these elements, volumetriclocking is very pronounced for materials such as Mises plasticity. The proviso plane strainis added here because volumetric locking will not occur in plane stress problems, for inplane stress the thickness of the element can change to accommodate incompressiblematerials.
The consequences of volumetric locking are almost a complete lack ofconvergence. In the presence of volumetric locking, displacements are underpredicted byfactors of 5 or more, so the results are completely worthless.Volumetric locking does not preclude the use of simplex elements for incompressiblematerials completely, for locking can be avoided by using special arrangements of theelements. For example, the cross-diagonal arrangement of triangles shown in Fig.??eliminates locking, Naagtegal et al.
However, meshing in this arrangement is similar tomeshing quadrilaterals, so the benefits arising from triangular and tetrahedral meshing arelost. In addition, this arrangement results in pressure oscillations, such as those describedsubsequently for quadrilaterals.When fully integrated, i.e. 2x2 Gauss quadrature for the quadrilateral, both the 4-nodequadrilateral and the hexahedron lock for incompressible materials.
Volumetric locking canbe eliminated in these elements by using reduced integration, namely one-point quadrature,or selective-reduced integration, which consists of one-point quadrature on the volumetricterms and 2x2 quadrature on the deviatoric terms; this is described in detail later. Theresulting quadrilateral will then exhibit good convergence properties in the displacements.However, the element still is plagued by one flaw: it exhibits pressure oscillations due tothe failure of the quadrilateral with modified quadrature to satisfy the BB-condition, whichis described later. As a consequence, the pressure field will often be oscillatory, with apattern of pressures as shown in fig, ??.
This oscillatory pattern in the pressures is oftenknown as checkerboarding. Checkerboarding is sometimes harmless: for example, inmaterials governed by the Mises law the response is independent of pressure, so pressureoscillations are not very harmful, although they lead to errors in the elastic strains.Checkerboarding can also be eliminated by filtering procedures. Nevertheless it isundesirable, and a user of finite elements should at least be aware of its possibility withthese elements. Pressure oscillations also occur for the mixed elements based on multifield variational principles. In fact mixed elements are in many cases identical or verysimilar in performance to selective reduced integration elements, since theoretically they arein many cases equivalent, Malkus and Hughes().
Some stabilization procedures for BBoscillations have been developed; they are described and discussed in Section ??.In large scale computations, the fastest form of the quadrilateral and hexahedron is theone-point quadrature element: it is often 3 to 4 times as fast as the selective-reducedquadrature quadrilateral element. In three dimensions, the speedup is of the order of 6 to 8.9The one-point quadrature element also suffers from pressure oscillations, and in additionpossess instabilities in the displacement field. These instabilities are shown in Fig. ??, andhave various names: hourglassing, keystoning, kinematic modes, spurious zero energymodes and chickenwiring are some of the appellations for these modes.
The control ofthese modes has been the topic of considerable research, and they can be controlled quiteeffectively. In fact, the rate of convergence is not decreased by a consistent control of thesemodes, so for many large scale calculations, one-point quadrature with hourglass controlare very effective.
Hourglass control is described in Sections ???.The next highest order elements are the 6-node triangle and the 8 and 9 nodequadrilaterals. The counterparts in three dimensions are the 10 node tetrahedron and the 20and 27 node quadrilaterals. The 6-node triangle and 9-node quadrilateral have a completequadratic displacement field and complete linear strain field when the edges of the elementare straight. Ciarlet and Raviart() in a landmark paper proved that the convergence of theseelements is quadratic when the displacement of the midside nodes is small compared to thelength of the elements; whether the distortions introduced by a mesh in normal meshgeneration are small is often an open question.
These elements satisfy the quadratic andlinear patch tests when the element sides are straight, but only the linear patch test when theelement sides are curved. In other words, these elements cannot reproduce a quadraticdisplacement field exactly when the sides are not straight. Of course, curved sides are anintrinsic advantage of finite elements, for they enable boundary conditions to be met forhigher order elements, but curved sides should only be used for exterior surfaces, sincetheir presence decreases the accuracy of the element.
In nonlinear problems with largedeformations, the performance of these elements degrades when the midside nodes movesubstantially; this had already been discussed in the one-dimensional context in Example2.8.2. Element distortion is a pervasive difficulty in the use of higher order elements forlarge-deformation analysis: the convergence rate of higher order elements degradessignificantly as they are distorted, and in addition solution procedures often fail whendistortion becomes excessive.The 6-node triangle does not lock for incompressible materials, but it fails the BBpressure stability test for incompressible materials.
The 9-node quadrilateral whendeveloped appropriately by a mixed variational principle with a linear pressure fieldsatisfies the pressure stability test and does not lock. It is the only element we havediscussed so far which has flawless behavior for incompressible materials.In summary, element technology deals with two major quirks:1. volumetric locking, which prevents convergence for incompressible and nearlyincompressible materials;2.
pressure oscillations which result from the failure to meet the BB condition.For low-order elements, the presence of one of these flaws is nearly unavoidable. Thequadrilateral with reduced integration and a pressure stabilization or pressure filter appearsto be the best of the low-order elements. When speed of computation is a consideration, astabilized quadrilateral with one-point quadrature appears to be optimal. Only the 9-nodequadrilateral and 27-node hexahedral are flawless elements for imcompressible materials,and the fact that no flaws have been discovered so far does not preclude that none will everbe discovered. Almost all of these difficulties are driven by incompressibility, and persist10for near-incompressiblity. When the material is compressible, or when considering twodimensional plane stress, standard element procedure can be used.Error Norms. In order to compare these elements further, it is worthwhile to study aconvergence theorem which has been proven for linear problems.
Although this theoremhas not been proven for the nonlinear regime, it provides insight into element accuracy.For the purpose of studying this convergence theorem, we will first define some normsfrequently used in error analysis of finite elements. These will also be used to evaluatesome of the element technology developed later in this Chapter.Errors in finite element analysis are measured by norms. A norm in functionalanalysis is just a way of measuring the distance between two functions.
A norm of thedifference between a finite element solution and the exact solution to a problem is a measureof the error in the solution. The most common norms for the evaluating the error in a finiteelement solution are the L2 norm and the error in energy. The L2 norm of a vector functionfi ( x ) is defined by2fi ( x ) 0 = fi ( x ) fi ( x) dΩ Ω1∫(8.2.1)where the subscript nought on the symbol for the norm designates the L2 norm.
It can beseen that the L2 norm is always positive, and measures an average or mean value of thefunction. To use the L2 norm for a measure of error for a finite element solution, wedenote the finite element solution for the displacement by u h ( x ) and the exact solution byu( x ) . The error in the finite element solution at any point can then be expressed by thevector e (x ) = uh (x ) − u (x ) . Since we seek a single number for the error, we will use themagnitude of the vector e (x ) . which is e (x ) ⋅ e( x) .
Thus we can define the error in thedisplacements by the L2 norm as2= e (x )⋅ e( x) dΩΩ1e (x ) 0∫2hh= u − u ⋅ u − u dΩΩ1oru − uh0∫()()(8.2.2)This error norm measures the average error in the displacements over the domain of theproblem. There are many other error norms. We have chosen to use this one because themost powerful and most well known results are expressed in terms of this norm.Furthermore, it gives a measure of error which is useful for engineering purposes.The second norm we will consider are the norms in Hilbert space. The H1 norm of avector function is defined by112fi ( x ) 1 = fi ( x ) fi ( x) + fi, j (x ) fi, j (x ) dΩ Ω1∫(8.2.3)This norm is a good measure of the error in the derivatives of a function. It includes theterms from the L2 norm given in Eq.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
