Augarde, C. E.

Ullah, Zahur

English

The element free Galerkin method (EFGM) [1] is one of the most robust meshless methods for the solution of elasto-statics problems. In the EFGM, moving least squares (MLS) shape functions are used for the approximation of the field variable. The essential boundary conditions cannot be implemented directly as in the case of Finite Element Method (FEM), because the MLS shape functions do not possess the Kronecker-delta property and use Lagrange multipliers instead. In this paper the recently developed local maximum entropy shape functions are used in the EFGM for the approximation of the field variable instead of MLS. As the local maximum entropy shape functions possess the Kronecker-delta property at the boundaries so the essential boundary conditions are enforced directly as in the case of FEM. Two benchmark problems, a cantilever beam subjected to parabolic traction at the free end and an infinite plate with circular hole subjected to unidirectional tension are solved to show the implementation and performance of the current approach. The displacement and stresses calculated by the current approach show good agreement with the analytical results

Augarde, Charles.E

Queen's University Belfast Research Portal

Solution of elasto-statics problems using the element-free Galerkin method with local maximum entropy shape functions

Augarde, Charles E

Durham Research Online

The element free Galerkin method (EFGM) [1] is one of the most robust meshless methods for the solution of elasto-statics problems. In the EFGM, moving least squares (MLS) shape functions are used for the approximation of the field variable. The essential boundary conditions cannot be implemented directly as in the case of Finite Element Method (FEM), because the MLS shape functions do not possess the Kronecker-delta property and use Lagrange multipliers instead. In this paper the recently developed local maximum entropy shape functions are used in the EFGM for the approximation of the field variable instead of MLS. As the local maximum entropy shape functions possess the Kroneckerdelta property at the boundaries so the essential boundary conditions are enforced directly as in the case of FEM. Two benchmark problems, a cantilever beam subjected to parabolic traction at the free end and an infinite plate with circular hole subjected to unidirectional tension are solved to show the implementation and performance of the current approach. The displacement and stresses calculated by the current approach show good agreement with the analytical results

Ulster University's Research Portal

18th UK Conference on Computational Mechanics (ACME-UK)29th–31st March 2010SouthamptonSolution of elasto-statics problems using the element free Galerkin methodwith local maximum entropy shape functionsZahur Ullah1 Charles Augarde21 School of Engineeringand Computing SciencesDurham UniversitySouth Road, DurhamDH1 3LE, UKzahur.ullah@dur.ac.uk2 School of Engineeringand Computing SciencesDurham UniversitySouth Road, DurhamDH1 3LE, UKcharles.augarde@dur.ac.ukwww.dur.ac.uk/charles.augardeKey Words: meshless methods, EFGM, moving least squares, maximum entropy shape functions.ABSTRACTThe element free Galerkin method (EFGM) [1] is one of the most robust meshless methods for thesolution of elasto-statics problems. In the EFGM, moving least squares (MLS) shape functions are usedfor the approximation of the field variable. The essential boundary conditions cannot be implementeddirectly as in the case of Finite Element Method (FEM), because the MLS shape functions do notpossess the Kronecker-delta property and use Lagrange multipliers instead. In this paper the recentlydeveloped local maximum entropy shape functions are used in the EFGM for the approximation of thefield variable instead of MLS. As the local maximum entropy shape functions possess the Kronecker-delta property at the boundaries so the essential boundary conditions are enforced directly as in thecase of FEM. Two benchmark problems, a cantilever beam subjected to parabolic traction at the freeend and an infinite plate with circular hole subjected to unidirectional tension are solved to show theimplementation and performance of the current approach. The displacement and stresses calculated bythe current approach show good agreement with the analytical results.1 INTRODUCTIONThe finite element method (FEM) is the most prominent tool for the solution of boundary value prob-lems in solid mechanics. However there are certain classes of problems for which the FEM is not anideal choice, e.g. crack growth and large deformation problems, since elements can become distortedaffecting solution accuracy and remeshing is required at different stages, which is computationally veryexpensive. Meshless methods are therefore a suitable choice for these problems, because only a set ofnodes is required for the problem discretization. However, to compete with the FEM some technicalproblems mentioned in the literature must be addressed. One of the main being the imposition of theessential boundary conditions. The EFGM in [1] uses moving least squares (MLS) shape functionsfor the approximation of the field variable. These shape functions do not possess the Kronecker-deltaproperty and essential boundary condition are imposed using the method of Lagrange multipliers. Thedimension of the final system of equations is increased and the stiffness matrix is no longer positivedefinite [2]. Here the EFGM is reformulated using the local maximum entropy shape functions recentlypresented in [3]. As the local maximum entropy shape functions possess the Kronecker-delta propertyat the boundary so essential boundary conditions can be implemented directly as in the FEM.2 MAXIMUM ENTROPY SHAPE FUNCTIONConsider mutually independent events x1, x2, ..., xn within a sample space Ω with unknown probabili-ties p1, p2, ..., pn respectively. The quantity for measuring the amount of information or uncertainty ofthe finite scheme is termed as information entropy [4] and is given asH(p1, ...., pn) = −N∑i=1pi log pi. (1)The most likely probability distribution with constraintsn∑i=1pi = 1 andn∑i=1pigr (xi) = 〈gr(x)〉, where〈gr(x)〉 is known as the expectation of gr(x) can be determined using Jaynes’ principle of maximumentropy.Maximize [H (p1, p2, ..., pn)] = −n∑i=1pi log pi. (2)We can consider the probabilities above to be the unknown shape functions we wish to determinewhich will, however, be highly non-local and non-interpolating (termed global maximum entropy shapefunctions). The local maximum entropy shape function formulation is summarized in [3] asH(p,m) = −n∑i=1pi log(pimi). (3)Here mi is the prior distribution, which is used in the calculation of pi. Using the analogy betweenprobabilities and shape functions and using the above strategymaximize(H(φ,w) = −n∑i=1φi log(φiwi)), (4)subject to the linear reproducing constraintsn∑i=1φi = 1,n∑i=1φix˜i = 0,n∑i=1φiy˜i = 0. (5)Where wi is the prior distribution, i.e. any weight function, and φi are the shape functions. The shapefunctions are found by the method of Lagrange multipliers 1φi =ZiZ, Zi = wie−λ1x˜i−λ2y˜i , Z =n∑j=1Zj . (6)Here λ1 and λ2 are the Lagrange multipliers determined using the dual formulation, i.e. to minimize FF = logZ (λ1, λ2) . (7)F is a convex function and Newton’s method is used to solve Equation (7). The expression for thederivatives of the shape functions is given as∇φi = φi(∇fi −n∑i=1φi∇fi)(8)where∇fi = ∇wiwi+ λ+ x˜i[H−1 −H−1A] , A = n∑k=1φkx˜k⊗∇wkwk. (9)H is the Hessian matrix and ⊗ is the dyadic product of two vectors.3 IMPLEMENTATIONTo demonstrate the use of these shape functions in the EFGM two benchmark problems are now pre-sented. A cantilever beam subjected to parabolic traction at the free end and an infinite plate withcircular hole subjected to unidirectional tension.1Not the same use as for essential boundary conditionsyu xuXYDLFigure 1: Geometry and coordinate system the cantilever beam problem3.1 Cantilever BeamThe behaviour of a cantilever beam subjected to parabolic traction at the free end is examined. Thegeometry, coordinate system and boundary conditions for the problem, which are more complicatedthan is often appreciated, are given in Figure 1. The exact solutions for the displacement and stressfields are given in [5] asu (x, y) =Py6EI[(6L− 3x)x+ (2 + ν) y2 − 3D22(1 + ν)], (10a)v (x, y) = − P6EI[3νy2 (L− x) + (3L− x)x2] (10b)andσxx =P (L− x) yI, σyy = 0, σxy = − P2I[D24− y2]. (11)E is the modulus of elasticity, ν is the Poisson’s ratio and I is the second moment of area. The problemis solved for the plane stress case with P = 1000, ν = 0.3, E = 3 × 107, D = 12, L = 48 and unitthickness, all in compatible units. 99 (11 × 9) nodes, 40 (10 × 4) background cells and 4 × 4 gaussquadrature per cell is used, while 4 gauss quadrature per line cell is used for the integration of the forceon the traction boundary. Figure: 2(a) shows the normal stress σxx and Figure: 2(b) shows the shearstress σxy at x = L/2 vs y. The numerical solution for the stresses can be seen to be almost the sameas the exact solution.−6 −4 −2 0 2 4 6−1500−1000−50005001000Yσxx  EFG with MXTExact(a)−6 −4 −2 0 2 4 6−140−120−100−80−60−40−20020Yσxy  EFG with MXTExact(b)Figure 2: Normal and shear stress at x=L/2 for the cantilever beam problem3.2 Infinite Plate with a circular holeConsider an infinite plate with circular hole of radius 1 in the centre . Due to symmetry only a portion ofthe upper right quadrant is modelled as shown in Figure 3. The exact solution for the stress field is givenin [5]. In this analysis a plane stress condition is assumed with E = 1000 and ν = 0.25. 121 (11× 11)nodes and 100 (10×10) background cells with (4×4) gauss quadrature per cell and 4 gauss quadratureper line cell is used for the integration on the traction boundary. Figure 4(a) shows σxx vs y while Figure4(b) shows σyy vs x. Once again the numerical solutions match closely the analytical solutions.yy2 0t xy1 4YX0xu 0yu  1 0t xxFigure 3: Geometry and coordinate system the infinite plate with a hole problem1 1.5 2 2.5 3 3.5 4 4.5 511.522.533.5Yσxx  EFG with MXTExact(a)1 1.5 2 2.5 3 3.5 4 4.5 5−1−0.8−0.6−0.4−0.200.20.4Xσyy  EFG with MXTExact(b)Figure 4: Normal stress at x=0 and at y=0 for the infinite plate with a hole problem4 CONCLUSIONIn this paper the recently developed local maximum entropy shape functions are used in the EFGM.These allow the imposition of essential boundary conditions directly as in the FEM. The two bench-mark problems, a cantilever beam subjected to parabolic traction at the free end and an infinite platewith a circular hole subjected to unidirectional tensile load are analyzed by the current approach. Thenumerical results shows good agreement with the analytical solutions.References[1] T. Belytschko, Y.Y. Lu, and L. Gu. Element-free Galerkin methods. International Journal forNumerical Methods in Engineering, 37:227–256, 1994.[2] S. Ferna´ndez-Me´ndez and A. Huerta. Imposing essential boundary conditions in mesh-free meth-ods. Computer Methods in Applied Mechanics and Engineering, 193:1257–1275, 2004.[3] N. Sukumar and R.W. Wright. Overview and construction of meshfree basis functions: from mov-ing least squares to entropy approximants. International Journal for Numerical Methods in Engi-neering, 70:181–205, 2007.[4] A.I. Khinchin. Mathematical Foundation of Information Theory. Dover Publications, Inc. , NewYork, 1957.[5] S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd edition,1970.

Solution of Elasto-Statics Problems Using the Element-Free Galerkin Method with Local Maximum Entropy Shape Functions

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Solution of Elasto-Statics Problems Using the Element-Free Galerkin Method with Local Maximum Entropy Shape Functions

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