Belytschko T. - Introduction (779635), страница 89
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(1), but when applied to real approximations, theerrors in the derivatives dwarf the errors in the function itself, so they play an insignificantrole. If we take the H1 norm of the error in the displacements, i.e. by lettingfi ( x ) = uih (x ) − ui ( x) we otain a useful measure of the error in strains. The errors in norm,incidentally, are usually similar to the error in energy, Hughes(,p.273) which is defined by2a (u, u) 1 = eij Cijkl ekl dΩeij = εij − εijhΩ1∫(8.2.4)Note the similarity of this form to the strain energy defined in Eq. ().
In the aboveexpression, the error in strain replaces the strain in Eq.(), hence the name energy error instrain.We conclude with a few more facts on norms. The Hr norm is generated in terms ofthe rth derivatives of the function. Thus the H0 norm is equivalent to the L2 norm, Eq (1),whereas the H2 norm would involve the squares of the second derivatives. These normsexist, i.e. the integral corresponding to the norms Hr, is integrable, when the function is ofcontinuity Cr-1 . This can be seen quite easily for the H1 norm: if the function fi ( x ) is notC0, i.e.
if it is discontinuious, then the derivatives will be Dirac delta functions at the pointsof the discontinuity. The square of a Dirac delta function cannot be integrated, so the normcan not be evaluated. The kinematic admissibility conditions () are often stated in terms ofHilbert spaces, so in Eq () the requirement could be replaced by .
The latter is often foundin the literature, but we used the simpler concept since we were not concerned withconvergence proofs. For more on norms, seminorms, and other good stuff of this type seeHughes(), Oden and Reddy() or Strang().Convergence Results for Linear Problems. The fundamental convergence results forlinear finite elements is given in the following.
If the finite element solution is generated byelements which can reproduce polynomials of order k, and if the solution u( x ) issufficiently smooth for the Hilbert norm Hr to exist, thenu − uhm≤ Chα ur,α =min (k +1 −m,r − m )(8.2.5)where h is a measure of element size and C is an arbitrary constant which is independent ofh and varies from problem to problem.12We will now examine the implications of this theorem for various elements.
Theparameter α indicates the rate of convergence of the finite element solution: the greater thevalue of α , the faster the finite element solution converges to the exact solution andtherefore the more accurate the element. It is important to note that the rate of convergenceis limited by the smoothness of the solution in space.
An elastic solution is analytic, i.e.infinitely smooth, if there are no acute corners or cracks, so in that case r tends towardinfinity. Therefore, the second term in the definition for α , r − m , plays no role forsmooth solutions. However, if the solution is not very smooth, i.e. if there arediscontinuities in the derivatives or higher order derivatives, then r is finite. For example,if there are discontinuities in the second derivatives, then r is at most 2, and the second termplays a role.We first examine what Eq. (5) means for smooth elastic solutions for various elementsfor the error in displacements.
In that case, we consider the H0 norm, which is equivalentto the L2 norm, so m = 0 . The 3-node triangle, the 4-node quadrilateral, the 4-nodetetrahedron and the 8-node hexahedron all reproduce in linear polynomials exactly; thisresult is proven for the isoparametric elements in Section 8.?; therefore k = 1 . Therefore,for the elements with linear completeness just listed we obtain thatα = min(k +1− m, r− m ) =min (1 +1− 0, ∞− 0) = 2This result is illustrated in Fig. (), which shows a log-log plot of the error for the platewith a hole; details of this problem are given in Section ??. In a log-log plot, the graph oferror in displacements versus element size according to Eq.
(5) is given by a straight linewith a slope α = 2 . The solution in this case is said to converge quadratically. The actualnumerical results compare with this theoretical result quite closely, although the slopedeviates 5% or so from the theoretical result. Equation 5 is an aymptotic result whichshould hold only as the element size goes to zero, but remarkably it agrees very well withnumerical experiments with realistic meshes.We next consider the higher order elements, namely the 6-node triangle, the 9-nodequadrilateral, the tetrahedron and the 27-node hexahedron with straight edges.
In this casek = 2 , and for an elastic solution the remaining constants are unchanged. We find then thatα = 3 , so the rate of convergence is cubic in the displacements. This increase of one orderin convergence is quite significant, as illustrated in the results shown in Fig. ??. In effect,the choice of a higher order element here buys a tremendous amount of accuracy.The results for the strains, i.e. the derivatives of the displacement field, are similar.
Inthis case m = 1 since the error in strains is indicated by the H1-norm. The rates ofconvergence are then one order lower, α = 1 for elements with linear completeness, k = 1 ,and α = 2 for elements with quadratic completeness, k = 2 . the results are illustrated for aplate with a hole in fig. ??.Convergence in Nonlinear Problems. The behavior of elements for nonlinear problemsaccording to this theorem, Eq (), will depend on the smoothness of the constitutiveequation. If the constitutive equation is very smooth, such as hyperelastic models forrubber, then the rate of convergence are expected to be the same as for elastic, linear13materials. However, for constitutive equations which are not smooth, such as elasticplastic materials, the second term in the definition of α governs the accuracy.
Forexample in an elastic-plastic material, the relation between stress and strain is C0.Therefore the displacements are at most C1, and r = 2 . It can now be seen that the rate ofconvergence of the displacements now is at most of order 2, i.e. α = 2 , regardless ofwhether the completeness of the element is as indicated by k is linear or quadratic. Thusthere appears to be no benefit in going to a higher order element for these materials.Similarly, the rate of convergence in the strains is at most of order α = 1 .
Thus, if theconstitutive equation is not very smooth, the benefits of a higher order element can be lost.In summary, for smooth constitutive equations, higher order elements areadvantageous because of their higher rate of convergence. If the constitutive equation lackssufficient smoothness, then there is no advantage in going to higher order elements. Theseresults also are relevant for dynamic problems: when the signals are very smooth, there issome benefit in going to higher order elements, provided that a consistent mass matrix isused. For signals which lack smoothness, there is little advantage to higher orderelements.These statements do not take cognizance of the deterioration of element performancewith large deformations.
When the deformations are so large that the elements are highlydistorted, then the accuracy of the higher order elements also decreases. These provisospertain to both total and updated Lagrangian meshes, but not to Eulerian meshes. Thus, theamount of element distortion expected should also be considered in the choice of an elementfor nonlinear analysis.It could be argued that even elastic problems in practical situations have discontinuitiesin derivatives due to different materials. However, in linear problems, the element edgesare usually aligned with the material interfaces.
In that case, the full accuracy of higherorder elements can be retained since they can model discontinuities in derivatives effectivelyalong element edges. In elastic-plastic problems, on the other hand, discontinuities floatthrough the model and as the problem evolves, they proliferate. Thus their effects innonlinear problems are more devastating to accuracy.It should be stressed that the convergence results (5) has only been proven for linearproblems. However, the major impediment to obtaining such convergence results fornonlinear problems is probably the lack of stability of nonlinear solutions.
It is likely thatthe estimates given above, which are based on interpolation error estimates, play a similarrole in nonlinear problems. This conjecture appears to be verified by numericalconvergence studies which have verified that the estimates () apply in nonlinear problemsquite well.8.3. The Patch TestsThe patch tests are an extremely useful for examining the soundness of elementformulations, for examining their stability and convergence behavior, and checking theproper implementation of an element in a compute program. The patch test was firstconceived by Irons() to examine the soundness of a nonconforming plate element. In thisoriginal form, the patch test was primarily a test for polynomial completeness, i.e.
theability to reproduce exactly a polynomial of order k. It has been proposed by Strang() that14the patch test is necessary and sufficient for convergence. Subsequently, the patch test hasbeen generalized and modified so that it can test also for stability in pressures anddisplacements, simo(), bathe(). Methods for implementing the patch test in explicitprograms have also been developed, Belytschko(). Special versions of the patch test to testperformance in large displacement analysis can also be constructed.
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