Belytschko T. - Introduction (779635), страница 90
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In the following wedescribe these various forms of the patch test.Before describing the patch test, it is worthwhile to define a few terms and point out afew overlaps in terms which are at times confusing. In functional analysis, the termcompleteness refers to the ability of an approximation to approximate a function arbitrarilyclosely. A sequence of functions φI ( x) is complete in Hr if for any function f ( x ) ∈H r ,f (x) −n∑ a IφI ( x)I=1→ 0 as n →∞(8.2.3)rThus any set of functions is complete if it can approximate any function of a specifiedcontinuity arbitrarily closely, when the error is measured by an appropriate norm. Theappropriate norm is any norm which exists or a lower order norm.In the preceding we have referred to polynomial completeness.
A better terms whichhas emerged in wavelet thoery is the reproducing condition. The reproducing condition isdefined by the ability of an approximation to reproduce a function exactly. Thus for aninterpolant such as a finite element shape function, the reproducing conditions state that ifthe nodal values of an element are given by pi (x J ) where pi (x ) is an arbitrary function,thenm∑ N J (x ) pi (x J ) = pi( x )(8.3.4)J =1This equation is quite subtle and contains more than first meets the eye. It states that whenthe reproducing condition holds, the shape functions or interpolants are able to exactlyreproduce the given function pi (x ) . For example, if the shape functions are able toreproduce the constant and linear functions, then we havemmJ =1J =1∑ N J (x ) = 1, ∑ NJ ( x )x jJ = x j(8.3.5)This is called linear completeness by Hughes(), but the term reproducing condition seemsmore appropriate, since completeness refers to a more general condition described by (4).Therefore, when using completeness in the sense of Hughes we will use the termpolynomial completeness.Any approximation which satisfies the linear reproducing conditions can be shown tobe complete.
On the other hand, the converse does not hold. Consider for example theFourier series: they are complete, but they cannot reproduce a linear polynomial.15A third definition pertinent to convergence is the defintion of consistency.Consistency is usually defined in the context of finite difference methods. According to thestandard definitions of consiostency, a discretization in space D (u ) is a consistentapproximation of a partial differential equation L ( u) = 0 if the error is on the order of themeshsize, i.e. if( )L ( u) − D (u ) = o hn ,n ≥1The above states that the truncation error must tend to zero as the nodal spacing, i.e.
theelement size tends to zero. For time dependent problems, the disretization error will be afunction of the time step and the element size h, and the truncation error will depend onboth. For a time-independent problem in one dimensionStandard Patch Test. We first describe the standard patch test which checks forpolynomial completeness of the displacement field, i.e. the ability of the element toreproduce polynomilas of a specified order. In addition, the test can be used to check theoverall implementation of the element in the program; sometimes the shape functions arecorrect, but the element in a program fails the patch test anyway because of faultyprogramming.In the standard patch test, a patch of elements such as shown in Fig.?? is used.
Theelements should be distorted as shown because the behavior of distorted elements isimportant and can differ from that of regular elements. No body forces should be applied,and the material properties should be uniform and linear elastic in the patch. Thedisplacements of the nodes on the periphery of the patch are then prescribed according tothe order of the patch test.
For a linear patch test in two dimensions, the displacement fieldis given byux = a1 x +a2 x x + a3x yuy = a1y + a2 y x + a3 y ywhere a Ii are constants set by the user; they should all be nonzero to test the reproducingcondition completely. This displacement field is used to set the prescribed displacements ofthe nodes on the periphery of the patch, so the prescribed displacements areuIx = a1 x +a2 x x I + a3x y IuIy = a1y + a2 y x I + a3 y yITo satisfy the patch test, the finite element solution should be given by () throughoutthe patch: the nodal displacements at the interior nodes should be given by () and the strainsshould be constant and given by the application of the strain-displacement equations to thedisplacement in ():ε x = ux , x = a2 x , ε y = u y, y = a3 y2ε xy = ux , y + uy , x = a3 x + a2y16The stresses should similarly be constant and correspond to what would be obtainedby multiplying the above strains in the elastic, linear law used in the program.
All of theseconditions should be met to a high degree of precision, on the order of the precision in thecomputer used.That rationale for the standard patch test are the reproducing condition and the fact that() corresponds to an exact solution to the governing equations for linear elasticity. It caneasily be seen that () is an exact solution to the elastic problem. Since the strains areconstant, and the material properties uniform, the stresses are constant. Therefore, sincethere are no body forces, the equilibrium equation () is satisfied exactly.
Since linear elasticsolutions are unique, Eq. () then represents a unique solution to the equations. If the finiteelement procedure is able to reproduce the linear field, it should be able to replicate thissolution exactly because the trial functions include this solution!When the patch test fails, then the finite element is either not complete, i.e.
it can notreproduce the linear field exactly, or there is an error in the program in developing thediscrete equations or in the solution of the discrete equations. Whether the reproducingconditions are satisfied can be checked independently by setting the nodal displacementsaccording to () at all nodes and then checking the strains at all quadrature points. This testin fact suffices as a test of the reproducing conditions, and hence of convergence of theelement.
Going through the solution procedure is primarily a check on the program.Patch Test in Explicit Programs. The patch test as applied above is not readilyapplicable to explicit programs because these program do not have a means for solving thelinear static equations. However, the patch test can be modified for use in explicitprograms as described in elytschko and Chiang(). The basic idea is to prescribe the intialvelocities by a linear field identical to Eq.(), soHere aij are arbitrary constant values, but they should be very small because in mostprograms the geometric nonlinearities will be triggered otherwise. The program is thenused to integrate the equations of motion one time step; no extrenal forces should be appliedand a linear, hypoelastic material model such as Eq.
() should be used. The rate-ofdeformation or the strains and the accelerations at the end of time step are then checked.The rate-of-deformation should have the correct constant values in all of the elements andthe accelerations should vanish at all of the interior nodes. The accelerations should vanishbecause the stresses should aslo be constant and according to the momentum equation, inthe absence of body forces, the accelerations should vanish.The test should be met to a high degree of precision if the constants aij are smallenough. For example, when the constants aij are of order , the accelerations should not belarger than order .Patch Tests for Stability. Simo, Taylor and Z have devised a modified patch test withthe aim of checking for stability, primarily in the stability of the displacement field ratherthan the instability of the pressures.
It can also test whether the program treats tractionboundary conditions exactly. The main difference from the standard patch test is that thedisplacements are not prescribed at all nodes. Instead, displacement boundary conditions17are prescribed only for the minimal number of components needed to prevent rigid bodyrotation. An example of the test is shown in Fig.
??.This test is not an infallible test for detecting instabilities. Furthermore, it can onlydetect displacement instabilities, not pressure instabilities. To thoroughly check an elementfor displacement instabilities, it is also worthwhile to to an eigenvalue analysis on a singlefree element, i.e. a completely unconstrained element. The number of zero eigenvaluesshould be equal to the number of rigid body modes. For example, in two dimensionalanalysis, an element or a patch of elements should posess three zero eigenvalues, which arearise forom two translations and on erotation, whereas in three dimensions, an elementshould posses six zero eigenvalues, three translational and three rotational rigid bodymodes.
If there are more zero eigenvalues, this indicates an element which may exhibitdisplacement instabilities; this characteristic is also called rank deficiency of the stiffnessmatrix, as discussed in Sectiopn 8.?.8.6. Isoparametric Element 4-Node QuadrilateralIn this Section, the isoparametric elements are developed in two dimensions, with anemphasis on the 4-node quadrilateral. The objective is to present a setting in which we canexplain some of the concepts described in the preceding Sections. The displacement fieldfor QUAD4 is given by4ux ξ,η = ∑ NI ξ,η uxII=14uy ξ,η = ∑ NI ξ,η uyI(8.2.1)I=1where NI is the isoparametric shape function for node I given byNI ξ,η = 1 1 + ξIξ 1 + ηIη(1.2.2)4uxI and uyI are the displacements at node I, and u x (ξ,η) and uy (ξ,η) give the displacementfield within the element domain.
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