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Finite Element Method - Point - Based approximations - Element - free glaerkin - and other meshlees methords_16 The description of the laws of physics for space- and time-dependent problems are usually expressed in terms of partial differential equations (PDEs). For the vast majority of geometries and problems, these PDEs cannot be solved with analytical methods. Instead, an approximation of the equations can be constructed, typically based upon different types of discretizations. These discretization methods approximate the PDEs with numerical model equations, which can be solved using numerical methods. The solution to the numerical model equations are, in turn, an approximation of the real solution to the PDEs. The finite element method (FEM) is used to compute such approximations.
16 Point-based approximations; element-free Galerkin - and other meshless methods 16.1 Introduction In all of the preceding chapters, the finite element method was characterized by the subdivision of the total domain of the problem into a set of subdomains called elements The union of such elements gave the total domain The subdivision of the domain into such components is of course laborious and difficult necessitating complex mesh generation Further if adaptivity processes are used, generally large areas of the problem have to be remeshed For this reason, much attention has been given to devising approximation methods which are based on points without necessity of forming elements When we discussed the matter of generalized finite element processes in Chapter 3, we noted that point collocation or in general finite differences did in fact satisfy the requirement of the pointwise definition However the early finite differences were always based on a regular arrangement of nodes which severely limited their applications To overcome this difficulty, since the late 1960s the proponents of the finite difference method have worked on establishing the possibility of finite difference calculus being based on an arbitrary disposition of collocation points Here the work of Girault,’ Pavlin and Perrone,* and Snell et d 3should be mentioned However a full realization of the possibilities was finally offered by Liszka and Orkis~,~,’ and Krok and Orkisz6 who introduced the use of least square methods to determine the appropriate shape functions At this stage Orkisz and coworkers realized not only that collocation methods could be used but also the full finite element, weak formulation could be adopted by performing integration Questions of course arose as to what areas such integration should be applied Liszka and Orkisz4 suggested determining a ‘tributary area’ to each node providing these nodes were triangulated as shown in Fig 16.1(a) On the other hand in a somewhat different context Nay and Utku7 also used the least square approximation including triangular vertices and points of other triangles placed outside a triangular element thus simply returning to the finite element concept We show this kind of approximation in Fig 16.1(b) Whichever form of tributary area was used the direct least square approximation centred at each node will lead to discontinuities of the function between the chosen integration areas and 430 Point-based approximations (4 Fig 16.1 Patches of triangular elements and tributary areas thus will violate the rules which we have imposed on the finite element method However it turns out that such rules could be violated and here the patch test will show that convergence is still preserved However the possibility of determining a completely compatible form of approximation existed This compatible form in which continuity of the function and of its slope if required and even higher derivatives could be accomplished by the use of so-called moving least square methods Such methods were originated in another context (Shepard,8 Lancaster and Salkauskas?”’) The use of such interpolation in the meshless approximation was first suggested by Nayroles et al,11-13 This formulation was named by the authors as the diflusefinite element method quickly realized the advantages offered by such an Belytschko and approach especially when dealing with the development of cracks and other problems for which standard elements presented difficulties His so-called ‘element-free Galerkin’ method led to many seminal publications which have been extensively used since An alternative use of moving least square procedures was suggested by Duarte and Oden.’62’7They introduced at the same time a concept of hierarchical forms by noting that all shape functions derived by least squares possess the partition of unity property (viz Chapter 8) Thus higher order interpolations could be added at each node rather than each element, and the procedures of element-free Galerkin or of the diffuse element method could be extended The use of all the above methods still, however, necessitates integration Now, however, this integration need not be carried out over complex areas A background grid for integration purposes has to be introduced though internal boundaries were no longer required Thus such numerical integration on regular grids is currently being used by B e l y t s c h k ~ ’ ~and ” ~ other approaches are being explored However an interesting possibility was suggested by BabuSka and Melenk.20>21 BabuSka and Melenk use a partition of unity but now the first set of basic shape functions is derived on the simplest element, say the linear triangle Most of the Function approximation 43 approximations then arise through addition of hierarchical variables centred at nodes We feel that this kind of approach which necessitates very few elements for integration purposes combines well the methodologies of ‘element free’ and ‘standard element’ approximation procedures We shall demonstrate a few examples later on the application of such methods which seem to present a very useful extension of the hierarchical approach Incidentally the procedures based on local elements also have the additional advantage that global functions can be introduced in addition to the basic ones to represent special phenomena, for instance the presence of a singularity or waves Both of these are important and the idea presented by this can be exploited In Volume 3, we shall show the application of this to certain wave phenomena, see Chapter 8, Volume T h s chapter will conclude with reference to other similar procedures which we not have time to discuss We shall refer to such procedures in the closure of this chapter 16.2 Function approximation We consider here a local set of n points in two (or three) dimensions defined by the coordinates xk,yk, z k ; k = 1,2, ,n or simply xk = [ x k , y kz,k ] at which a set of data values of the unknown function iik are given It is desired to fit a specified function form to the data points In order to make a fit it is necessary to: Specify the form of the functions, p ( x ) , to be used for the approximation Here as in the standard finite element method, it is essential to include low order polynomials necessary to model the highest derivatives contained in the differential equation or in the weak form approximation being used Certainly a complete linear and sometimes quadratic polynomial will always be necessary Define the procedure for establishing the fit Here we will consider some least squarefit methods as the basis for performing the fit The functions will mostly be assumed to be polynomials, however, in addition other functions can be considered if these are known to model well the solution expected (e.g., see Chapter 8, Volume on use of ‘wave’ functions) 16.2.1 Least square fit We shall first consider a least square fit scheme which minimizes the square of the distance between n data values iik defined at the points xk and an approximating function evaluated at the same points fi(xk).We assume the approximation function is given by a set of monomials pi n C(X) = pi(x)aj = p(x)a (16.1) j= in which p is a set of linearly independent polynomial functions and a is a set of parameters to be determined A least square scheme is introduced to perform the 432 Point-based approximations fit and this is written as (see Chapter 14 for similar operations): Minimize n J = c ( i i ( x k ) - iik)2= (16.2) k= where the minimization is to be performed with respect to the values of a Substituting the values of at the points xk we obtain (16.3) where This set of equations may be written in a compact matrix form as (16.4) where Pk = P(Xk) We can define the result of the sums as (16.5) (16.6) in which P= ["] and "=( 'l} Pn Un The above process yields the set of linear algebraic equations Ha = g = PTU which, provided H is non-singular, has the solution a = H-'g = H-'PTU (16.7) We can now write the approximation for the function as li = P(X) H-'PTU = N(x)U where N(x) are the appropriate shape or basis functions In general Ni(q) is not unity as it always has been in standard finite element shape functions However, the partition of unity [viz Eq (8.4)] is always preserved provided p(x) contains a constant Example: Fit of a linear polynomial To make the process clear we first consider a dataset, iik, defined at four points, xk, to which we desire to fit an approximation given by a linear polynomial + C(x) = a1 x a + y a = p(x)a Function approximation 433 If we consider the set of data defined by xk = [ -4.0 -1.0 yk = [ iik = 5.0 -5.0 0.0 6.01 0.0 3.01 5.1 3.5 4.31 [-1.5 we can write the arrays as 1 -4 -1 -5 0 -1.5 5.1 3.5 4.3 and Using Eq (16.5) we obtain the values H=PTP= 5; ! ] [[ and g = P Tu- = { ::::} -20.1 which from Eq (16.7) has the solution a= { 3.1241 0.4745} -0.5237 Thus, the values for the least square fit at the data points are - 1A u= { g;:} 4.2820 The ast square fit for these data points is shown in Fig 16.2 and the difference between the data points and the values of the fit at x k is given in Table 16.1 16.2.2 Weighted least square fit Let us now assume that the point at the origin, xo = 0, is the point about which we are making the expansion and, therefore, the one where we would like to have the best accuracy Based on the linear approximation above we observe that the direct least square fit yields at the point in question the largest discrepancy In order to improve the fit we can modify our least square fit for weighting the data in a way that emphasizes the effect of distance from a chosen point We can write such a weighted least square f i t as the minimization of ( 16.8) where w is the weighting function Many choices may be made for the shape of the function w If we assume that the weight function depends on a radial distance, r , 434 Point-based approximations Fig 16.2 Least square fit: (a) four data points; (b) fit of linear function on the four data points Function approximation 435 Table 16.1 Difference between least square fit and data -4 S xk yk !k -1.500 uk -1.392 Difference -0.108 0 3.500 3.124 4.300 4.400 -1 -5 5.100 5.268 -0.168 0.376 -0.100 from the chosen point we have w = w(r); - r2 = (x - xo) (x - xo) One functional form for w(r) is the exponential Gauss function: w ( r ) = exp(-cr2); c > and r (16.9) For c = 0.125 this function has the shape shown in Fig 16.3 and when used with the previously given four data points yields the linear fit shown in Table 16.2 16.2.3 Interpolation domains and shape functions In what follows we shall invariably use the least square procedure to interpolate the unknown function in the vicinity of a particular node i The first problem is that when approximating to the function it is necessary to include a number of nodes equal at least to the number of parameters of a sought to represent a given polynomial This number, for instance, in two dimensions is three for linear polynomials and six for quadratic ones As always the number of nodal points has to be greater than or equal to the bare minimum which is the number of parameters required We should note in passing that it is always possible to develop a singularity in the equation used for solving a, i.e Eq (16.7) if the data points lie for instance on a straight line in two or three dimensions However in general we shall try to avoid such difficulties by reasonable spacing of nodes The domain of influence can well be defined by making sure that the weighting function is limited in extent so that any point lying beyond a certain distance r, are weighted by zero and therefore are not taken into account Commonly used weighting functions are, for instance, in direction r, given by which represents a truncated Gauss function Another alternative is to use a Hermitian interpolation function as employed for the beam example in Sec 2.10: w(r) = [ - ( k Y + ( ) ; Odrdr, (16.11) 436 Point-based approximations Fig 16.3 Weighting function for Eq (16.9): c = 0.125 or alternatively the function 4-1 = {I' (k7ln ; Odrdr, ; r>r, and n 2 - (16.12) is simple and has been effectively used For circular domains, or spherical ones in three dimensions, a simple limitation of r, suffices as shown in Fig 16.4(a) However occasionally use of rectangular or hexahedral subdomains is useful as also shown in that figure and now of course the weighting function takes on a different form: Odxdx,; {:(x)yI(y); W(X,Y) = ; X Odydy,; and i , j > > Xm > Y > Y m with [ (:7] X,(x)= - - ;; y , ( y ) = [I - ( ; ] j Table 16.2 Difference between weighted least square fit and data xk yk -4 tk uk - 1s o Error -0.620 -0.880 -1 -5 5.100 5.247 -0.148 0 3.500 3.4872 4.300 5.246 0.013 -0.946 (16.13) Function approximation 437 Fig 16.4 Two-dimensional interpolation domains: (a) circular; (b) rectangular The above two possibilities are shown in Fig 16.4 Extensions to three dimensions using these methods is straightforward Clearly the domains defined by the weighting functions will overlap and it is necessary if any of the integral procedures are used such as the Galerkin method to avoid such an overlap by defining the areas of integration We have suggested a couple of possible ideas in Fig 16.1 but other limitations are clearly possible In Fig 16.5, we show an approximation to a series of points sampled in one dimension The weighting function here always embraces three or four nodes Limiting however the domains of their validity to a distance which is close to each of the points provides a unique definition of interpolation The reader will observe that this interpolation is Piecewise least sauare aooroximation Fig 16.5 A one-dimensionalapproximation to a set of data points using parabolic interpolation and direct least square fit to adjacent points 438 Point-based approximations discontinuous We have already pointed out such a discontinuity in Chapter 3, but if strictly finite difference approximations are used this does not matter It can however have serious consequences if integral procedures are used and for this reason it is convenient to introduce a modification to the definition of weighting and method of calculation of the shape function which is given in the next section 16.3 Moving least square approximations - restoration of continuity of approximation The method of moving least squares was introduced in the late 1960s by Shepard' as a means of generating a smooth surface interpolating between various specified point values The procedure was later extended for the same reasons by Lancaster and Salkauska~~ to~deal ' ~ with very general surface generation problems but again it was not at that time considered of importance in finite elements Clearly in the present context the method of moving least squares could be used to replace the local least squares we have so far considered and make the approximation fully continuous In moving least square methods, the weighted least square approximation is applied in exactly the same manner as we have discussed in the preceding section but is established for every point at which the interpolation is to be evaluated The result of course completely smooths the weighting functions used and it also presents smooth derivatives noting of course that such derivatives will depend on the locally specified polynomial To describe the method, we again consider the problem of fitting an approximation to a set of data items Ui, i = 1, ,n defined at the n points xi.We again assume the approximating function is described by the relation m u(x) z ti(.) = C p j ( x ) a j = p(x)a (16.14) j= I where pi are a set of linearly independent (polynomial) functions and aiare unknown quantities to be determined by the fit algorithm A generalization to the weighted least square fit given by Eq (16.8) may be defined for each point x in the domain by solving the problem n w,(xk - x)[iik- p(xk)ul2= J(X) = (16.15) k= I In this form the weighting function is defined for every point in the domain and thus can be considered as translating or moving as shown in Fig 16.6 This produces a continuous interpolation throughout the whole domain Figure 16.7 illustrates the problem previously presented in Fig 16.5 now showing continuous interpolation We should note that it is now no longer necessary to specify 'domains of influence' as the shape functions are defined in the whole domain The main difficulty with this form is the generation of a moving weight function which can change size continuously to match any given distribution of points xk with a limited number of points entering each calculation One expedient method Galerkin weighting and finite volume methods 453 The approximate solution to forms given by Eq (16.68) may be achieved using moving least squares and alternative methods for performing the domain integrals 16.6.2 Subdomain collocation - finite volume method A simple extension of the point collocation method is to use subdomains (elements) defined by the Voronoi neighbour criterion The integrals for each subdomain are approximated as a constant evaluated at the originating point as nd + nb E(vi)TF(u;)ri= c(~i)~D(ui)R, i (16.69) I + where nd nb = n, the total number of unknown parameters appearing in the approximations of u and v The validity of the above approximation form can be established using patch tests (see Chapter 10) This approach is often called subdomain collocation or thefinite volume method This approach has been used extensively in constructing approximations for fluid flow problem^.^^-^' It has also been employed with some success in the solution of problems in structural mechanic^.^' 16.6.3 Galerkin methods - diffuse elements Moving least square approximations have been used with weak forms to construct Galerkin type approximations The origin of this approach can be traced to the work of L i ~ z k and a ~ ~O r k i ~ z 'Additional ~ work, originally called the diffuse element approximation, was presented in the early 1990s by Nayroles et ~ f " - 'Beginning ~ in the mid-1990s the method has been extensively developed and improved by Belytschko and coauthors under the name element-free Galerkin.'41'5142~43 A similar procedure, call 'hp-clouds', was also presented by Oden and Duarte 16317,44 Each of the methods is also said to be 'meshless', however, in order to implement a true Galerkin process it is necessary to carry out integrations over the domain What distinguishes each of the above processes is the manner in which these integrations are carried out In the element-free Galerkin method a background 'grid' is often used to define the integrals whereas in the hp cloud method circular subdomains are employed Differing weights are also used as means to generate the moving least square approximation The interested reader is referred to the appropriate literature for more details Another source to consult for implementation of the EFG method is reference 19 Here we present only a simple implementation for solution of an ordinary differential equation Example: Galerkin solution of ordinary differential equations The moving least square approximation described in Sec 16.3; is now used as a Galerkin method to solve a second-order ordinary differential equation For an arbitrary function W ( x ) , a weak form for the differential equation may be deduced using the procedures 454 Point-based approximations presented in Chapter Accordingly, we obtain (16.70) subject to the boundary conditions u(0) = g l and u(L) = g Using a hierarchical moving least square form a p-order polynomial approximation to the dependent variable may be written as n ii(x) = fl(x)qjp(x)iq (16.71) j= where qjp =[1 x-xj (x-Xj) (X-xjy] (16.72) Note that in the above form we have used the representation qjp(x)uP = iij + q(x)bj The approximation to the weight function is similarly taken as n @(x) = fl(x)qjp(x)w; (16.73) j= in which W; are arbitrary parameters satisfying W ( )= W ( L )= The approximation yields the discrete problem = e ( W y ) T/ ; q ; @ f I= (16.74) (x) dx Since W: is arbitrary, the solution to the approximate weak form yields the set of equations = j I q ; @ f (x) dx; i = , , , n (16.75) The set of equations only needs to be modified to satisfy the essential boundary equations This is accomplished by replacing the equations corresponding to W , = W, = by iil = g l and ii, = g The Galerkin form requires only first derivatives of the approximating functions as opposed to the second derivatives required for the point collocation method This reduction, however, is accompanied by a need to perform integrals over the domain For weighting functions given by Eq (16.12) all functions entering the approximation are polynomial and rational polynomial expressions, thus, a closed form evaluation is impractical Accordingly, we evaluate integrals using Gauss and Fig 16.14 String on elastic foundation solution: 3-point Gauss quadrature Fig 16.15 String on elastic foundation solution: 4-point Gauss quadrature 456 Point-based approximations Fig 16.16 String on elastic foundation solution: 4-point Gauss-Lobatto quadrature Fig 16.17 String on elastic foundation solution: 5-point Gauss-Lobatto quadrature Use of hierarchic and special functions based on standard finite elements 457 Gauss-Lobatto quadrature over each interval generated by the basis points in the moving least square representation (i.e., xj for j = 1,2, ,n) As an example of the type of solutions possible we consider the string on elastic foundation problem given in the previous section For the parameters a = 0.004, c = with loading f = -1 and zero boundary conditions a Galerkin solution using and point Gauss quadrature and and point Gauss-Lobatto quadrature is shown in Figs 16.14-16.17 A mesh consisting of nine equally spaced points is used to define the intervals for the solution and quadrature The weight function is generated for k = 0, p = with a span of 2.1 mesh points Based upon this elementary example it is evident that the answers for a nine-point mesh depend on accurate evaluation of integrals to produce high-quality answers 16.7 Use of hierarchic and special functions based on standard finite elements satisfying the partition of unity requirement 16.7.1 Introduction In Sec 16.4,we discussed the possibility of introducing hierarchical variables to shape functions based on moving least square interpolations However a simpler approach to hierarchical forms and indeed to extensions by other functions can be based on simple finite element shape functions One important application of the partition of unity method starts from a set of finite element basis functions, Ni(x).An approximation to u ( x ) is now given by u(x) M ti(x) = Ni(x)[tii L + &(x)bai] (16.76) a where Ni(x)is the conventional (possibly isoparametric) finite element shape function at node i, q i ) are global functions associated with node i, and tii, and bai are parameters associated with the added global hierarchical functions We must note that as before Ui will not represent a local value of the function unless the function q’ become zero at the node i Here we assume that conventional shape functions which satisfy the partition of unity condition CNi=l (16.77) i are used Thus, the above form is a hierarchic finite element method based on the partition of We note in particular that the function qg) may be different for each node and thus the form may be effectively used in an adaptive finite element procedure as described in Chapter 15 Equation (16.76) provides options for a wide choice of functions for q$: Polynomial functions In this case the method becomes an alternative hierarchical scheme to that presented in Part of Chapter 458 Point-based approximations Harmonic 'wave' functions This is a multiscale method and will be discussed in detail in Volume 3 Singular functions These can be used to introduce re-entrant corner or singular load effects in elliptic problems (e.g., heat conduction or elasticity forms) Derivatives of Eq (16.76) are computed directly as (16.78) The reader will note that the narrow band structure of the standard finite element method will always be maintained as it is determined by the connectivity of Ni.Note also that the standard element on which the shape functions Niwere generated can be used for all subsequent integrations Such a formulation is very easy to fit into any finite element program 16.7.2 Polvnomial hierarchical method To give more details of the above hierarchical finite element method we first consider the one-dimensional approximation in a two-noded element where in which and (16.80) We recall that N + N2 = and N l x l + N2x2 = x Investigation of the term xk in the approximation = N1 ( x ) [GI + xjakl] + N2(x)[G2 + x'ak2] (16.81) we observe that a linear dependence with the usual finite element approximation occurs when iii = x i i and k = with bll = b I 2= bl In this case Eq (16.81) becomes ii = [NIX1 - = xbo + N2X2lbO + [Nl + N21Xbl + Xbl- (16.82) In one dimension linear dependence can be avoided by setting k to in Eqs (16.79) and (16.80) However, in two and three dimensional problems the linear dependence cannot be completely avoided, and we address this n e ~ t ~ ~ ' ~ ' An approximation over two-dimensional triangles may be expressed as + u ( x , y ) M i i ( x , y )= CLj[iij ~ ( ~ ) b i ] i= (16.83) Use of hierarchic and special functions based on standard finite elements 459 where Li are the area coordinates defined in Chapter We consider the case where complete quadratic functions are added as Y2 I (16.84) to give a complete second-order polynomial approximation for u Although this gives a complete second-order polynomial approximation there are two ways in which the cubic term x2y can be obtained The first sets iii = bil = bi3 = and bi2 = xi& giving The second alternative to compute the same term sets ii.I = b.12 - b.13 - and bil = yi& giving li = ELi [x2].y;& = x 2y& i= A similar construction may be made for the polynomial term xy2 An alternative is to construct the interpolation to depend on each node as (16.85) This form, while conceptually the same as the original formulation, appears to be better conditioned and also avoids some of the problems of linear d e p e n d e n ~ yIn ~~ Sec 16.7.4 we will discuss in more detail a methodology to deal with the problem of linear dependency, however, before doing so we illustrate the use of the hierarchical finite element method by an application to two-dimensional problems in linear elasticity 16.7.3 Application to linear elasticity In the previous section the form for polynomial interpolation in two dimensions was given Here we consider the use of the interpolation to model the behaviour of problems in linear elasticity For simplicity only the displacement model for plane strain as discussed in Chapters and is considered; however, the use of the hierarchic interpolations can easily be extended to other forms and to mixed models For a displacement model the finite element arrays may be computed using Eq (2.24) For two-dimensional plane strain problems, the strain-displacement 460 Point-based approximations - au - - ax E= au - (16.86) aY au au -+- a y axInserting the interpolations for u and u given by Eq (16.76) and using Eq (16.78) to compute derivatives, the strain-displacement relations become N E=C i= 8Nf aY - 0 i= + [E] ( g q ; Nik aqk ) aY (16.87) The first term is identical to the usual finite element strain-displacement matrices [see Eq (4.10b)l and the second term has identical structure to the usual arrays Thus, the development of all element arrays follows standard procedures A quadratic triangular element For a triangular element with linear interpolation the shape functions and quadratic polynomial hierarchic terms are given by N i = Li and Eq (16.85), respectively Using isoparametric concepts the coordinates are given by (16.88) and are used to construct all polynomials appearing in hierarchical form (16.85) A set of patch tests is first performed to assess the stability and consistency of the above hierarchic form The set consists of one, two, four, and eight element patches as shown in Fig 16.18 First, we perform a stability assessment by determining the number of zero eigenvalues for each patch The results for hierarchical interpolation are shown in Table 16.3 The eigenproblem assessment reveals that the hierarchic interpolation has excess zero eigenvalues (i.e., spurious zero energy modes) only for meshes consisting of Use of hierarchic and special functions based on standard finite elements 461 Fig 16.18 Patches for eigenproblem assessment one or two elements Furthermore, only two element meshes in which one side is a straight line through both elements have excess zero values Once the mesh has no straight intersections the number of zero modes becomes correct (e.g., contain only the three rigid body modes) Consistency tests verify that all meshes contain terms of up to quadratic polynomial order - thus also validating the correctness of the coding As a simple test problem using the hierarchical finite element method we consider a finite width strip containing a circular hole with diameter half the width of the strip The strip is subjected to axial extension in the vertical direction and, due to symmetry Table 16.3 Triangle element patch tests: Number of zero eigenvalues, minimum non-zero value, and maximum value ( k = 2) - quadratic hierarchical terms Mesh No zero Min value 2a 2b 2c I 5 3 4.7340E 01 4.0689E + 01 4.1971E + 02 S728E 02 1.0446E + 02 9.5560E + 01 + + Max value + 2.0560E 06 2.1543E + 05 2.2648E + 05 2.3883E 06 2.9027E + 05 3.4813E + 05 + 462 Point-based approximations Fig 16.19 Hierarchic elements: tension strip Fig 16.20 lsoparametric six-noded elements: tension strip Use of hierarchic and special functions based on standard finite elements 463 Table 16.4 Hierarchical element Boundary segments straight ~~ ~ ~ Nodes Elements Equations Energy 30 85 279 1003 28 112 448 1792 156 537 1971 7527 131.7088 127.8260 126.7641 126.5908 of the loading and geometry, only one quadrant is discretized as shown in Figs 16.19 and 16.20 The meshes in Fig 16.19 employ the hierarchical interpolation considered above; whereas those in Fig 16.20 use standard six-node isoparametric quadratic triangles with two degrees of freedom per node (Le., u and w) The material is taken as linear elastic with E = 1000 and v = 0.25 The half-width of the strip is 10 units and the half-height is 18 units The hole has radius The problem size and computed energy (which indicates solution accuracy) are shown in Table 16.4 for the hierarchical method, in Table 16.5 for the six-node isoparametric formulation and in Table 16.6 for three-node linear triangular elements The six-node isoparametric method gives overall the best accuracy; however, the hierarchical element is considerably better than the three-node triangular element and offers great advantages when used in adaptive analysis.47 16.7.4 Solution of forms with linearly dependent equations A typical problem for a steady-state analysis in which the algebraic equations are generated from the hierarchical finite element form described above, such as given Table 16.5 Isoparametric element Boundary segments have curved sides Nodes Elements Equations Energy 30 279 1003 3795 28 112 448 1792 129 483 1863 731 127.3350 126.6483 126.5661 126.5593 Table 16.6 Linear triangular element Nodes Elements Equations Energy 30 85 279 1003 3795 28 112 448 1792 7168 36 129 483 1863 731 137.652 131.065 128.008 126.958 126.662 464 Point-based approximations by Eqs (16.83) and (16.84), produces algebraic equations in the standard form, i.e., Ka+f=O (16.89) where the parameters a include both nodal ili and hierarchical parameters bi We assume that occasionally the 'stiffness matrix' K and 'force' vector f include equations which are linearly dependent with other equations in the system and, thus, K can be singular If the system is solved by a direct elimination scheme (e.g., as described in Chapter or in books on linear algebra such as references 48 or 49) it is possible to set a tolerance for the pivot below which an equation is assumed to be linearly dependent and can be omitted from the calculations (e.g., see reference 50, 51) An alternative to the above is to perturb Eq (16.89) to [K + &DK]APk= f - KPk where DK are diagonal entries of K, E ( 16.90) is a specified value and ak+' = ak + Aak (16.91) is used to define an iterative strategy An initial guess of zero may be used to start the leads to rapid solution process Certainly a choice of a small value for E (e.g., convergence.47 16.8 Closure In this chapter we have considered a number of methods which eliminate or reduce our dependence on meshing the total domain There are a number of other approaches having the same aim which have been pursued with success These include the smooth particle hydrodynamics method (SPH) (Lucy,52Gingold and M ~ n a g h a n , ~ ~ B e n ~and ~ ~the ) reproducing kernel method (RPK) (Liu et d.55,56) applied to problems in solid and fluid mechanics Bonet and coworker^^^'^^ improve the method of SPH and show its possibilities Another approach has recently been introduced by Y a g a ~ a ~These ~ > ~are ' not described here and the reader is referred to the literature for details References V Girault Theory of a finite difference method on irregular networks SIAM J Num Anal., 1 , 260-82, 1974 V Pavlin and N Perrone Finite difference energy techniques for arbitrary meshes Comp Struct., 5,45-58, 1975 C Snell, D.G Vesey, and P Mullord The application of a general finite difference method to some boundary value problems Comp Struct., 13, 547-52, 1981 T Liszka and J Orkisz Finite difference methods of arbitrary irregular meshes in nonlinear problems of applied mechanics In Proc 4th Int Conference on Structural Mechanics in Reactor Technology, 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1997 ... square fit and data -4 S xk yk !k -1 .500 uk -1 .392 Difference -0 .108 0 3.500 3.124 4.300 4.400 -1 -5 5.100 5.268 -0 .168 0.376 -0 .100 from the chosen point we have w = w(r); - r2 = (x - xo) (x - xo)... 4 -point Gauss quadrature 456 Point- based approximations Fig 16.16 String on elastic foundation solution: 4 -point Gauss-Lobatto quadrature Fig 16.17 String on elastic foundation solution: 5 -point. .. is that approximations at points other 448 Point- based approximations Fig 16.9 Methods for selecting points: (a) cross; (b) Voronoi than those used to write the differential equations and boundary